Difference between revisions of "Tutorial 5 - 2010 - Solution/Question 2"

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.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715 \end{array}
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\
 
\Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715
 
\end{array}
 
  
 
*ITERATION 1*
 
*ITERATION 1*
Line 54: Line 50:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(1)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505)}{0.26}\\ T^{(1)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505)}{0.26}\\ T^{(1)} &= 179.8955\\ & \\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(179.8955\right) + 1.69\text{x}10^{-3}\left(179.8955\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(179.8955\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= -23255 \end{array}
T^{(1)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505)}{0.26}\\
 
T^{(1)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505)}{0.26}\\
 
T^{(1)} &= 179.8955\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(179.8955\right) + 1.69\text{x}10^{-3}\left(179.8955\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(179.8955\right)}\\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= -23255
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(1)} &= \left| \frac{\left(179.8955\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(1)} &= 0.5266 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23255\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(1)} &= 0.0128 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(1)} &= \left| \frac{\left(179.8955\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(1)} &= 0.5266 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23255\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(1)} &= 0.0128 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
 
*ITERATION 2*
 
*ITERATION 2*
Line 80: Line 60:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(2)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505)}{0.26}\\ T^{(2)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(179.8955)^{2} + \frac{1.5\text{x}10^{5}}{(179.8955)} - (-23505)}{0.26}\\ T^{(2)} &= 1140.4\\ & \\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(1140.4\right) + 1.69\text{x}10^{-3}\left(1140.4\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(1140.4\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= -22064 \end{array}
T^{(2)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505)}{0.26}\\
 
T^{(2)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(179.8955)^{2} + \frac{1.5\text{x}10^{5}}{(179.8955)} - (-23505)}{0.26}\\
 
T^{(2)} &= 1140.4\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(1140.4\right) + 1.69\text{x}10^{-3}\left(1140.4\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(1140.4\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= -22064
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(2)} &= \left| \frac{\left(1140.4\right)-\left(179.8955\right)}{\left(179.8955\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(2)} &= 5.3394 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-22064\right)-\left(-23255\right)}{\left(-23255\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(2)} &= 0.0512 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(2)} &= \left| \frac{\left(1140.4\right)-\left(179.8955\right)}{\left(179.8955\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(2)} &= 5.3394 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-22064\right)-\left(-23255\right)}{\left(-23255\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(2)} &= 0.0512 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
 
*ITERATION 3*
 
*ITERATION 3*
Line 106: Line 70:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(3)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505)}{0.26}\\ T^{(3)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(1140.4)^{2} + \frac{1.5\text{x}10^{5}}{(1140.4)} - (-23505)}{0.26}\\ T^{(2)} &= 6682.6\\ & \\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(6682.6\right) + 1.69\text{x}10^{-3}\left(6682.6\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(6682.6\right)}\\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= 49659 \end{array}
T^{(3)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505)}{0.26}\\
 
T^{(3)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(1140.4)^{2} + \frac{1.5\text{x}10^{5}}{(1140.4)} - (-23505)}{0.26}\\
 
T^{(2)} &= 6682.6\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(6682.6\right) + 1.69\text{x}10^{-3}\left(6682.6\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(6682.6\right)}\\
 
\Delta H_{r}^{0}\left(T^{(3)}\right) &= 49659
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(3)} &= \left| \frac{\left(6682.6\right)-\left(1140.4\right)}{\left(1140.4\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(3)} &= 4.8598 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\left(49659\right)-\left(-22064\right)}{\left(-22064\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(3)} &= 3.2507 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(3)} &= \left| \frac{\left(6682.6\right)-\left(1140.4\right)}{\left(1140.4\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(3)} &= 4.8598 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(3)} &= \left| \frac{\left(49659\right)-\left(-22064\right)}{\left(-22064\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(3)} &= 3.2507 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
  
Line 133: Line 81:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715 \end{array}
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\
 
\Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715
 
\end{array}
 
  
 
*ITERATION 1*
 
*ITERATION 1*
Line 143: Line 87:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(1)} &= T^{(0)} - 24097 - 0.26T^{(0)} + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505)\\ T^{(1)} &= (380) - 24097 - 0.26(380) + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505)\\ T^{(1)} &= 283.2531\\ & \\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(283.2531\right) + 1.69\text{x}10^{-3}\left(283.2531\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(283.2531\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= -23505.49146 \end{array}
T^{(1)} &= T^{(0)} - 24097 - 0.26T^{(0)} + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505)\\
 
T^{(1)} &= (380) - 24097 - 0.26(380) + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505)\\
 
T^{(1)} &= 283.2531\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(283.2531\right) + 1.69\text{x}10^{-3}\left(283.2531\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(283.2531\right)}\\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= -23505.49146
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(1)} &= \left| \frac{\left(283.2531\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(1)} &= 0.080 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23505.49146\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(1)} &= 0.0022 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(1)} &= \left| \frac{\left(283.2531\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(1)} &= 0.080 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23505.49146\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(1)} &= 0.0022 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
 
*ITERATION 2*
 
*ITERATION 2*
Line 169: Line 97:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(2)} &= T^{(1)} - 24097 - 0.26T^{(1)} + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505)\\ T^{(2)} &= (283.2531) - 24097 - 0.26(283.2531) + 1.69\text{x}10^{-3}(283.2531)^{2} + \frac{1.5\text{x}10^{5}}{(283.2531)} - (-23505)\\ T^{(2)} &= 282.7616\\ & \\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(282.7616\right) + 1.69\text{x}10^{-3}\left(282.7616\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(282.7616\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= -23504.913333 \end{array}
T^{(2)} &= T^{(1)} - 24097 - 0.26T^{(1)} + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505)\\
 
T^{(2)} &= (283.2531) - 24097 - 0.26(283.2531) + 1.69\text{x}10^{-3}(283.2531)^{2} + \frac{1.5\text{x}10^{5}}{(283.2531)} - (-23505)\\
 
T^{(2)} &= 282.7616\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(282.7616\right) + 1.69\text{x}10^{-3}\left(282.7616\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(282.7616\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= -23504.913333
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(2)} &= \left| \frac{\left(282.7616\right)-\left(283.2531\right)}{\left(283.2531\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(2)} &= 0.0017 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-23504.913333\right)-\left(-23505.49146\right)}{\left(-23505.49146\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(2)} &= 0.000026 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(2)} &= \left| \frac{\left(282.7616\right)-\left(283.2531\right)}{\left(283.2531\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(2)} &= 0.0017 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-23504.913333\right)-\left(-23505.49146\right)}{\left(-23505.49146\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(2)} &= 0.000026 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
 
*ITERATION 3*
 
*ITERATION 3*
Line 195: Line 107:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(3)} &= T^{(2)} - 24097 - 0.26T^{(2)} + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505)\\ T^{(3)} &= (282.7616) - 24097 - 0.26(282.7616) + 1.69\text{x}10^{-3}(282.7616)^{2} + \frac{1.5\text{x}10^{5}}{(282.7616)} - (-23505)\\ T^{(2)} &= 282.8483\\ & \\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(282.8483\right) + 1.69\text{x}10^{-3}\left(282.8483\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(282.8483\right)}\\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= -23505.0156 \end{array}
T^{(3)} &= T^{(2)} - 24097 - 0.26T^{(2)} + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505)\\
 
T^{(3)} &= (282.7616) - 24097 - 0.26(282.7616) + 1.69\text{x}10^{-3}(282.7616)^{2} + \frac{1.5\text{x}10^{5}}{(282.7616)} - (-23505)\\
 
T^{(2)} &= 282.8483\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(282.8483\right) + 1.69\text{x}10^{-3}\left(282.8483\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(282.8483\right)}\\
 
\Delta H_{r}^{0}\left(T^{(3)}\right) &= -23505.0156
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(3)} &= \left| \frac{\left(282.8483\right)-\left(282.7616\right)}{\left(282.7616\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(3)} &= 0.00031 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\left(-23505.0156\right)-\left(-23504.913333\right)}{\left(-23504.913333\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(3)} &= 0.0000044 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(3)} &= \left| \frac{\left(282.8483\right)-\left(282.7616\right)}{\left(282.7616\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(3)} &= 0.00031 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(3)} &= \left| \frac{\left(-23505.0156\right)-\left(-23504.913333\right)}{\left(-23504.913333\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(3)} &= 0.0000044 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
  
Line 222: Line 118:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715 \end{array}
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\
 
\Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715
 
\end{array}
 
  
 
*ITERATION 1*
 
*ITERATION 1*
Line 232: Line 124:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(1)} &= T^{(0)} - \left(- 24097 - 0.26T^{(0)} + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505) \right)\\ T^{(1)} &= (380) - \left(- 24097 - 0.26(380) + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505) \right)\\ T^{(1)} &= 432.0272\\ & \\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(432.0272\right) + 1.69\text{x}10^{-3}\left(432.0272\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(432.0272\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= -23547 \end{array}
T^{(1)} &= T^{(0)} - \left(- 24097 - 0.26T^{(0)} + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505) \right)\\
 
T^{(1)} &= (380) - \left(- 24097 - 0.26(380) + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505) \right)\\
 
T^{(1)} &= 432.0272\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(432.0272\right) + 1.69\text{x}10^{-3}\left(432.0272\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(432.0272\right)}\\
 
\Delta H_{r}^{0}\left(T^{(1)}\right) &= -23547
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(1)} &= \left| \frac{\left(432.0272\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(1)} &= 0.1369 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23547\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(1)} &= 0.00044 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(1)} &= \left| \frac{\left(432.0272\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(1)} &= 0.1369 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23547\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(1)} &= 0.00044 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
 
*ITERATION 2*
 
*ITERATION 2*
Line 258: Line 134:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(2)} &= T^{(1)} - \left(- 24097 - 0.26T^{(1)} + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505) \right)\\ T^{(2)} &= (432.0272) - \left(- 24097 - 0.26(432.0272) + 1.69\text{x}10^{-3}(432.0272)^{2} + \frac{1.5\text{x}10^{5}}{(432.0272)} - (-23505) \right)\\ T^{(2)} &= 473.7196\\ & \\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(473.7196\right) + 1.69\text{x}10^{-3}\left(473.7196\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(473.7196\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= -23524 \end{array}
T^{(2)} &= T^{(1)} - \left(- 24097 - 0.26T^{(1)} + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505) \right)\\
 
T^{(2)} &= (432.0272) - \left(- 24097 - 0.26(432.0272) + 1.69\text{x}10^{-3}(432.0272)^{2} + \frac{1.5\text{x}10^{5}}{(432.0272)} - (-23505) \right)\\
 
T^{(2)} &= 473.7196\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(473.7196\right) + 1.69\text{x}10^{-3}\left(473.7196\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(473.7196\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= -23524
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(2)} &= \left| \frac{\left(473.7196\right)-\left(432.0272\right)}{\left(432.0272\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(2)} &= 0.0965 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-23524\right)-\left(-23547\right)}{\left(-23547\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(2)} &= 0.00095 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(2)} &= \left| \frac{\left(473.7196\right)-\left(432.0272\right)}{\left(432.0272\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(2)} &= 0.0965 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-23524\right)-\left(-23547\right)}{\left(-23547\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(2)} &= 0.00095 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
 
*ITERATION 3*
 
*ITERATION 3*
Line 284: Line 144:
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} T^{(3)} &= T^{(2)} - \left(- 24097 - 0.26T^{(2)} + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505) \right)\\ T^{(3)} &= (473.7196) - 24097 - 0.26(473.7196) + 1.69\text{x}10^{-3}(473.7196)^{2} + \frac{1.5\text{x}10^{5}}{(473.7196)} - (-23505)\\ T^{(2)} &= 492.9904\\ & \\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(492.9904\right) + 1.69\text{x}10^{-3}\left(492.9904\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(492.9904\right)}\\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= -23510 \end{array}
T^{(3)} &= T^{(2)} - \left(- 24097 - 0.26T^{(2)} + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505) \right)\\
 
T^{(3)} &= (473.7196) - 24097 - 0.26(473.7196) + 1.69\text{x}10^{-3}(473.7196)^{2} + \frac{1.5\text{x}10^{5}}{(473.7196)} - (-23505)\\
 
T^{(2)} &= 492.9904\\
 
& \\
 
\Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\
 
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(492.9904\right) + 1.69\text{x}10^{-3}\left(492.9904\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(492.9904\right)}\\
 
\Delta H_{r}^{0}\left(T^{(3)}\right) &= -23510
 
\end{array}
 
  
 
.. math::
 
.. math::
 
 
\begin{array}{rl}
+
\begin{array}{rl} \epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(3)} &= \left| \frac{\left(492.9904\right)-\left(473.7196\right)}{\left(473.7196\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(3)} &= 0.0407 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\left(-23510\right)-\left(-23524\right)}{\left(-23524\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(3)} &= 0.00060 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}
\epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,x}^{(3)} &= \left| \frac{\left(492.9904\right)-\left(473.7196\right)}{\left(473.7196\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,x}^{(3)} &= 0.0407 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\
 
& \\
 
\epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\
 
\epsilon_{tol,f}^{(3)} &= \left| \frac{\left(-23510\right)-\left(-23524\right)}{\left(-23524\right)} \right| < \left(10^{-6}\right) ? \\
 
\epsilon_{tol,f}^{(3)} &= 0.00060 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.
 
\end{array}
 
  
  

Latest revision as of 03:09, 16 September 2018

← Question 1 (previous step) Back to all questions Next step: Question 3 →

Question 2 [1]

  1. Derive a \(g(x) = x\) function to use in the fixed-point algorithm.
  2. Show the first 3 iterations of using the fixed-point algorithm, starting with an initial guess of \(T = 380\) K.
  3. Will the fixed-point method converge for this problem, using your \(g(x)\)?

Solution

  1. The goal of fixed-point iterative methods is to locate "fixed-point(s)" associated with a given function. A fixed-point is special case of a function where in it maps a value \(x\) to itself (what this means in layman's terms is if you put a value \(x\) into a function you get \(x\) out, i.e. \(x = g(x)\)). Knowing that our objective is to find the root(s) of a known objective function (we could have just had a black box function...) we may then construct \(g(x)\) in way that gives it the greatest chance of converging. Let us start by looking at our objective function:

    \[f(T) = 0 = - 24097 - 0.26T + 1.69\text{x}10^{-3}T^{2} + \frac{1.5\text{x}10^{5}}{T} - (-23505)\]

    Likely the first choice of g(x) that popped to everyone's mind was

    \[T = \frac{- 24097 + 1.69\text{x}10^{-3}T^{2} + \frac{1.5\text{x}10^{5}}{T} - (-23505)}{0.26} = g(T)_{1}\]

    Given that we have a function \(f(T) = 0\), another good choice would be one of \(g(T) = T + f(T)\) or \(g(T) = T - f(T)\). Let's say we were trying to identify the lower of the two roots. In this case we would want \(f(T)\) to yield a negative value when \(T > T_{root}\) and a positive value when \(T < T_{root}\). Looking at the plot generated in Question 1 we see that the function is positive when \(T\) is smaller than the lower root and greater than the upper root and is negative when \(T\) is between the two roots. Therefore if we wished to locate the lower root \(g(T) = T - f(T)\) would locally satisfy the desired properties and if we wished to find the upper root \(g(T) = T + f(T)\) would locally satisfy the desired properties (Note that these are gross generalizations and do not prove stability or ensure convergence. They are merely educated guesses as to potentially good functions). Therefore we will also investigate both:

    \[g(T)_{2} = T + (- 24097 - 0.26T + 1.69\text{x}10^{-3}T^{2} + \frac{1.5\text{x}10^{5}}{T} - (-23505))\]
    \[g(T)_{3} = T - (- 24097 - 0.26T + 1.69\text{x}10^{-3}T^{2} + \frac{1.5\text{x}10^{5}}{T} - (-23505))\]
  2. Starting with a value of \(T^{(0)} = 380 K\):

    FUNCTION \(g(T)_{1}\)

    \[\begin{split}\begin{array}{rl} \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715 \end{array}\end{split}\]

    ITERATION 1

    \[\begin{split}\begin{array}{rl} T^{(1)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505)}{0.26}\\ T^{(1)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505)}{0.26}\\ T^{(1)} &= 179.8955\\ & \\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(179.8955\right) + 1.69\text{x}10^{-3}\left(179.8955\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(179.8955\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= -23255 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(1)} &= \left| \frac{\left(179.8955\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(1)} &= 0.5266 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23255\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(1)} &= 0.0128 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    ITERATION 2

    \[\begin{split}\begin{array}{rl} T^{(2)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505)}{0.26}\\ T^{(2)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(179.8955)^{2} + \frac{1.5\text{x}10^{5}}{(179.8955)} - (-23505)}{0.26}\\ T^{(2)} &= 1140.4\\ & \\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(1140.4\right) + 1.69\text{x}10^{-3}\left(1140.4\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(1140.4\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= -22064 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(2)} &= \left| \frac{\left(1140.4\right)-\left(179.8955\right)}{\left(179.8955\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(2)} &= 5.3394 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-22064\right)-\left(-23255\right)}{\left(-23255\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(2)} &= 0.0512 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    ITERATION 3

    \[\begin{split}\begin{array}{rl} T^{(3)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505)}{0.26}\\ T^{(3)} &= \frac{- 24097 + 1.69\text{x}10^{-3}(1140.4)^{2} + \frac{1.5\text{x}10^{5}}{(1140.4)} - (-23505)}{0.26}\\ T^{(2)} &= 6682.6\\ & \\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(6682.6\right) + 1.69\text{x}10^{-3}\left(6682.6\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(6682.6\right)}\\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= 49659 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(3)} &= \left| \frac{\left(6682.6\right)-\left(1140.4\right)}{\left(1140.4\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(3)} &= 4.8598 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\left(49659\right)-\left(-22064\right)}{\left(-22064\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(3)} &= 3.2507 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    FUNCTION \(g(T)_{2}\)

    \[\begin{split}\begin{array}{rl} \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715 \end{array}\end{split}\]

    ITERATION 1

    \[\begin{split}\begin{array}{rl} T^{(1)} &= T^{(0)} - 24097 - 0.26T^{(0)} + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505)\\ T^{(1)} &= (380) - 24097 - 0.26(380) + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505)\\ T^{(1)} &= 283.2531\\ & \\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(283.2531\right) + 1.69\text{x}10^{-3}\left(283.2531\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(283.2531\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= -23505.49146 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(1)} &= \left| \frac{\left(283.2531\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(1)} &= 0.080 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23505.49146\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(1)} &= 0.0022 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    ITERATION 2

    \[\begin{split}\begin{array}{rl} T^{(2)} &= T^{(1)} - 24097 - 0.26T^{(1)} + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505)\\ T^{(2)} &= (283.2531) - 24097 - 0.26(283.2531) + 1.69\text{x}10^{-3}(283.2531)^{2} + \frac{1.5\text{x}10^{5}}{(283.2531)} - (-23505)\\ T^{(2)} &= 282.7616\\ & \\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(282.7616\right) + 1.69\text{x}10^{-3}\left(282.7616\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(282.7616\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= -23504.913333 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(2)} &= \left| \frac{\left(282.7616\right)-\left(283.2531\right)}{\left(283.2531\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(2)} &= 0.0017 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-23504.913333\right)-\left(-23505.49146\right)}{\left(-23505.49146\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(2)} &= 0.000026 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    ITERATION 3

    \[\begin{split}\begin{array}{rl} T^{(3)} &= T^{(2)} - 24097 - 0.26T^{(2)} + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505)\\ T^{(3)} &= (282.7616) - 24097 - 0.26(282.7616) + 1.69\text{x}10^{-3}(282.7616)^{2} + \frac{1.5\text{x}10^{5}}{(282.7616)} - (-23505)\\ T^{(2)} &= 282.8483\\ & \\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(282.8483\right) + 1.69\text{x}10^{-3}\left(282.8483\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(282.8483\right)}\\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= -23505.0156 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(3)} &= \left| \frac{\left(282.8483\right)-\left(282.7616\right)}{\left(282.7616\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(3)} &= 0.00031 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\left(-23505.0156\right)-\left(-23504.913333\right)}{\left(-23504.913333\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(3)} &= 0.0000044 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    FUNCTION \(g(T)_{3}\)

    \[\begin{split}\begin{array}{rl} \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\ \Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715 \end{array}\end{split}\]

    ITERATION 1

    \[\begin{split}\begin{array}{rl} T^{(1)} &= T^{(0)} - \left(- 24097 - 0.26T^{(0)} + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505) \right)\\ T^{(1)} &= (380) - \left(- 24097 - 0.26(380) + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505) \right)\\ T^{(1)} &= 432.0272\\ & \\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(432.0272\right) + 1.69\text{x}10^{-3}\left(432.0272\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(432.0272\right)}\\ \Delta H_{r}^{0}\left(T^{(1)}\right) &= -23547 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(1)} &= \left| \frac{\left(432.0272\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(1)} &= 0.1369 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-23547\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(1)} &= 0.00044 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    ITERATION 2

    \[\begin{split}\begin{array}{rl} T^{(2)} &= T^{(1)} - \left(- 24097 - 0.26T^{(1)} + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505) \right)\\ T^{(2)} &= (432.0272) - \left(- 24097 - 0.26(432.0272) + 1.69\text{x}10^{-3}(432.0272)^{2} + \frac{1.5\text{x}10^{5}}{(432.0272)} - (-23505) \right)\\ T^{(2)} &= 473.7196\\ & \\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(473.7196\right) + 1.69\text{x}10^{-3}\left(473.7196\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(473.7196\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= -23524 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(2)} &= \left| \frac{\left(473.7196\right)-\left(432.0272\right)}{\left(432.0272\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(2)} &= 0.0965 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-23524\right)-\left(-23547\right)}{\left(-23547\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(2)} &= 0.00095 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]

    ITERATION 3

    \[\begin{split}\begin{array}{rl} T^{(3)} &= T^{(2)} - \left(- 24097 - 0.26T^{(2)} + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505) \right)\\ T^{(3)} &= (473.7196) - 24097 - 0.26(473.7196) + 1.69\text{x}10^{-3}(473.7196)^{2} + \frac{1.5\text{x}10^{5}}{(473.7196)} - (-23505)\\ T^{(2)} &= 492.9904\\ & \\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\ \Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(492.9904\right) + 1.69\text{x}10^{-3}\left(492.9904\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(492.9904\right)}\\ \Delta H_{r}^{0}\left(T^{(3)}\right) &= -23510 \end{array}\end{split}\]
    \[\begin{split}\begin{array}{rl} \epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(1)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,x}^{(3)} &= \left| \frac{\left(492.9904\right)-\left(473.7196\right)}{\left(473.7196\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,x}^{(3)} &= 0.0407 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\ & \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\ \epsilon_{tol,f}^{(3)} &= \left| \frac{\left(-23510\right)-\left(-23524\right)}{\left(-23524\right)} \right| < \left(10^{-6}\right) ? \\ \epsilon_{tol,f}^{(3)} &= 0.00060 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \end{array}\end{split}\]
  3. It would appear that \(g(T)_{1}\) is rapidly diverging while \(g(T)_{2}\) is converging to the lower root and \(g(T)_{3}\) is converging to the upper root.