Exercises

Exercises#

1. Linear fitting of the Van’t Hoff equation#

The Van’t Hoff equation relates the change in the equilibrium constant, \(K\), for a reaction, to a change in temperature, \(T\), given that the enthalpy change for the reaction, \(\Delta H\), is constant over the temperature range of interest.

\[\frac{\mathrm{d} \ln K}{\mathrm{d} T} = \frac{\Delta H}{RT^2}\]

This equation is derived from two standard expressions for the Gibbs free energy change for a reaction:

\[\Delta G = \Delta H - T \Delta S\]
\[\Delta G = -RT \ln K\]

Combining these two equations gives the linear form of the Van’t Hoff equation:

\[\ln K = -\frac{\Delta H}{RT} + \frac{\Delta S}{R}\]

Differentiating with respect to \(1/T\) (keeping \(\Delta H\) and \(\Delta S\) constant) gives the derivative form of the Van’t Hoff equation.

The linear form of the Van’t Hoff equation has the functional form of the equation for a straight line (hence “linear form”)

\[y = mx + c\]

Plotting experimental data as \(\ln K\) against \(1/T\) should, therefore, give an approximate straight line (as long as the enthalpy and entropy of reaction are approximately constant over the temperature range of interest). By fitting a straight line to these data, we can obtain “best fit” values for the slope and intercept, which correspond to \(\Delta H/R\) and to \(\Delta S/R\) in our model. By fitting to data we can, therefore, estimate \(\Delta H\) and \(\Delta S\) for a given reaction.

(a) The data file exercise_1.dat gives the following experimentally measured equilibrium constants at different temperatures for the reaction

\[2\mathrm{NO}_2 \rightleftharpoons 2\mathrm{N}_2\mathrm{O}_4\]

temperature (°C)

\(K\)

9

34.3

20

12

25

8.79

33

4.4

40

2.8

52

1.4

60

0.751

70

0.4

Read these data from the file exercise_1.dat into two numpy arrays.

(b) Convert your raw data into \(1/T\) (in inverse Kelvin) and \(\ln K\).

Plot these transformed data and confirm that they approximately show a linear relationship.

(c) Write a function called linear_model() with arguments \(x\), \(m\), and \(c\), that returns the corresponding \(y\) value for the function

\[y = mx + c\]

(d) Replot your raw \(1/T\) versus \(\ln K\) data and show that for \(m\) = 7400 and \(c\) = -22.5 your model approximately described your data.

(e) Write a function error_function() that takes as arguments a list of model parameters, a list of \(x\) values, and a list of observed \(y\) values. This function should call your linear_model() function to calculate the sum-of-squares error for a given pair of model parameters, \(m\) and \(c\).

\[\chi^2 = \sum_i\left[y_i - f(x_i)\right]^2\]

(f) Use scipy.optimize.minimize() to find the “best fit” values of \(m\) and \(c\) for your data. Use the previous values of \(m\) = 7400 and \(c\) = -22.5 as your starting guess.

(g) Replot your data, now including your “line of best fit”

(h) One way to perform linear regression in Python without writing your own model function and error function is to use scipy.stats.linregress():

from scipy.stats import linregress

result = linregress(x, y)

You can then access the slope and intercept as result.slope and result.intercept.

Use scipy.stats.linregress() to perform linear regression on your data, and confirm that you get the same best fit parameters, \(m\) and \(c\), as before.

2. Non-linear fitting of kinetic data#

The data in exercise_2.dat describe the decomposition of hydrogen peroxide in the presence of excess CeIII.

\[2\mathrm{H}_2\mathrm{O}_2 + 2\mathrm{Ce}^\mathrm{III} \longrightarrow 2\mathrm{H}_2\mathrm{O} + \mathrm{Ce}^\mathrm{III} + \mathrm{O}_2\]

Time (s)

[H2O2] / mol dm−3

2

6.23

4

4.84

6

3.76

8

3.20

10

2.60

12

2.16

14

1.85

16

1.49

18

1.27

20

1.01

(a) Read these data to numpy arrays, and plot time versus the concentration of H2O2.

(b) This reaction has a first-order rate equation, which will be our “model” for fitting the data:

\[[\mathrm{A}](t) = [\mathrm{A}]_0\exp(-kt)\]

Write a function first_order() to calculate this \([\mathrm{A}](t)\) as a function of \(t\), \([\mathrm{A}]_0\), and \(k\).

(c) Write a function error_function() that calculates the least-squares error for your first-order kinetic model compared to the experimental data.

(d) Use scipy.optimize.minimize() to find the best-fit values of \([\mathrm{A}]_0\), and \(k\). Use starting guesses of \(k\) = 0.1 and \(A\) = 7.

(e) Replot the raw data, plus the curve for your best-fit parameters.

(f) You can also perform non-linear least-squares fitting, without having to define an error function, using scipy.optimize.curve_fit().

from scipy.optimize import curve_fit

To use curve_fit(), you pass your model function, and the \(x\) and \(y\) data:

popt, pcov = curve_fit(model_function, x_data, y_data)

curve_fit() returns two numpy arrays. The first array contains the optimised model parameters. The second array contains a 2D array that describes the covariance of these parameters. The diagonal elements of this array are estimated uncertainties (standard deviations) for each model parameter, and the off-diagonal elements give information about the degree of correlation between the model parameters.

Use curve_fit() to confirm your best fit model parameters for your first-order kinetic model. You will need to pass in as arguments your model function, the observed \(x\) data (time), and the observed \(y\) data (H2O2 concentration).