https://codingforchemistsbook.com/book_material/chapter-8/index.html Data There is no data required for Chapter 8. Alternatively, individual files can be found in the Data section. Code from chapter ''' A code to numerically solve a reaction mechanism that is irreversible but has two intermediates using solve_ivp from scipy.integrate Written by: Ben Lear and Chris Johnson (authors@codechembook.com) v1.0.0 - 250217 - initial version ''' import scipy.integrate as spi from codechembook.quickPlots import quickScatter # First we need to define a function containing our differential rate laws. def TwoIntermediatesDE(t, y, k): ''' Differential rate laws for a catalytic reaction with two intermediates: A + cat ->(k1) X X + B ->(k2) Y Y ->(k3) C + cat REQUIRED PARAMETERS: t (float): the cutrent time in the simulation. Not explicitly used but needed by solve_ivp y (list of float): the current concentration k (list of float): the rate RETURNS: (list of float): rate of change of concentration in the order of y ''' A, B, cat, X, Y, C = y # unpack concentrations to convenient variables k1, k2, k3 = k # unpack rates dAdt = -k1 * A * cat dBdt = -k2 * X * B dcatdt = -k1 * A * cat + k3 * Y dXdt = k1 * A * cat - k2 * X * B dYdt = k2 * X * B - k3 * Y dCdt = k3 * Y return [dAdt, dBdt, dcatdt, dXdt, dYdt, dCdt] # Set up initial conditions and simulation parameters y0 = [1.0, 1.0, 0.2, 0.0, 0.0, 0.0] # concetrations (mM) [A, B, cat, X, Y, C] k = [5e1, 1e1, 5e0] # rate (1/s) time = [0, 10] # simulation start and end times (s) # Invoke solve_ivp and store the result object solution = spi.solve_ivp(TwoIntermediatesDE, time, y0, args = [k]) # Plot the results quickScatter(x = solution.t, # need a list of six identical time axes y = solution.y, name = ['[A]', '[B]', '[cat]', '[X]', '[Y]', '[C]'], xlabel = 'Time (s)', ylabel = 'Concentration (mM)', output = 'png') ''' A code to numerically solve a first order kinetics problem using solve_ivp from scipy.integrate Written by: Ben Lear and Chris Johnson (authors@codechembook.com) v1.0.0 - 250217 - initial version ''' import numpy as np import scipy.integrate as spi from plotly.subplots import make_subplots # First we need to define a function containing our differential rate law. def FirstOrderDE(t, y, k): ''' Differential rate law for first-order kinetics REQUIRED PARAMETERS: t (float): the cutrent time in the simulation. Not explicitly used but needed by solve_ivp y (float): the current concentration k (float): the rate Returns: (float): the rate of change of concentration with time ''' dydt = -k * y return dydt # Set up initial conditions and simulation parameters y0 = 1.0 # concetration (mM) k = 1.0 # rate (1/s) time = [0, 10] # simulation start and end times (s) # Invoke solve_ivp and store the result object as solution solution = spi.solve_ivp(FirstOrderDE, time, [y0], args = [k]) # Compute the known analytical solution at each simulation time point anal_y = np.exp(-1 * k * solution.t) # Plot the numerical and analytical solutions and the difference between them fig = make_subplots(2, 1) fig.add_scatter(x = solution.t, y = solution.y[0], mode = 'markers', name = 'Numerical') # The data from the numerical solution fig.add_scatter(x = solution.t, y = np.exp(-1 * k * solution.t), mode = 'lines', name = 'Exact') # The corresponding analytical solution fig.add_scatter(x = solution.t, y = 100 * (solution.y[0] - anal_y) / anal_y, name = 'Error', row = 2, col = 1) # The percent difference between them # Update the plot appearance fig.update_xaxes(title = 'Time (s)') fig.update_yaxes(title = 'Concentration (mM)', row = 1) fig.update_yaxes(title = 'Percent Difference', row = 2) fig.update_layout(template = 'simple_white') fig.show('png') Solutions to Exercises Targeted exercises Simulating the kinetics of one elementary reaction using scipy.solve ivp Exercise 0 Working from the code presented in Section Simulating_the_kinetics_of_one_elementary_reaction_using_scipy.solve_ivp, write a script that plots the maximum error in the simulation versus the relative tolerance specification within scipy.solve_ivp(). You can use the rtol keyword to set this. Use a range of rtol values that span $10^{-1}$ to $10^{-9}$ changing by factors of 10. Hugo en-us authors@codingforchemistsbook.com (Benjamin Lear and Christopher Johnson) authors@codingforchemistsbook.com (Benjamin Lear and Christopher Johnson)