# Collection and analysis of rate data - 2013

 Class date(s): 28 February
• F2011: Chapter 7
• F2006: Chapter 5

### 28 February 2013 (07C)

R code for the example covered in class:

t <- c(0, 50, 100, 150, 200, 250)    # [minutes]
# or you can write t <- seq(0, 250, 50)

CA <- c(154, 114, 87, 76, 70, 58)  # [mol/m^3]

# Plot the data first. See the expected trend. It it decreasing nonlinearly, so it is either first or second order (but not zeroth order)
plot(t, CA)
grid()

# Assume it is a first order, rA = -k CA
# So plot log(CA0/CA) against t
plot(t, log(CA/CA))
grid()

# Does not quite look like a straight line, but let us fit a least-squares model anyway

mod.first <- lm( log(CA / CA) ~ t)
summary(mod.first)
#Call:
#lm(formula = log(CA/CA) ~ t)
#
#Residuals:
#       1        2        3        4        5        6
#-0.09425  0.02134  0.10646  0.05647 -0.04646 -0.04357
#
#Coefficients:
#            Estimate Std. Error t value Pr(>|t|)
#(Intercept) 0.0942475  0.0604645   1.559 0.194066
#t           0.0037033  0.0003994   9.272 0.000753 ***
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
#Residual standard error: 0.08354 on 4 degrees of freedom
#Multiple R-squared: 0.9555,	Adjusted R-squared: 0.9444

# Now assume a second order system, rA = -k CA^2
# So plot 1/CA against t
plot(t, 1/CA)
grid()

# Definitely more linear. Let us fit the least squares model to check:
mod.second <- lm( I(1/CA) ~ t)
summary(mod.second)
#Call:
#lm(formula = I(1/CA) ~ t)
#
#Residuals:
#         1          2          3          4          5          6
#-2.751e-04 -5.219e-05  6.146e-04  2.227e-04 -7.051e-04  1.951e-04
#
#Coefficients:
#             Estimate Std. Error t value Pr(>|t|)
#(Intercept) 6.769e-03  3.692e-04   18.33 5.21e-05 ***
#t           4.111e-05  2.439e-06   16.86 7.26e-05 ***
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
#Residual standard error: 0.0005101 on 4 degrees of freedom
#Multiple R-squared: 0.9861,	Adjusted R-squared: 0.9826
#F-statistic: 284.2 on 1 and 4 DF,  p-value: 7.259e-05


So we got a higher $$R^2$$ on the second order model. The reaction rate can be considered second order. What is the reaction rate constant, $$k_A$$ for this system?