Contents
Entropy of CO2 mixing
Colin Ophus - 2025 Jan 2
This notebook plots the entropy of mixing a closed isothermal system consisting of two chambers at equal pressure:
- A volume fraction α of CO2
- A volume fraction of other gases
The entropy of mixing is equal to
We will also plot the contributions from each individual side of the system, and the specific entropy for CO2 in both general terms and for 1 atm / 300 K conditions, which are equal to
import numpy as np
import matplotlib.pyplot as plt# Define x*log(x) function
def x_log_x(x):
# f(x) = x * log(x)
f = np.zeros_like(x)
sub = np.logical_and(
x > 0.0,
x < 1.0,
)
f[sub] = x[sub] * np.log(x[sub])
return f
# Calculate entropy of mixing
alpha = np.linspace(0.0,1.0,1001)
s_mix_co2 = -x_log_x(alpha)
s_mix_other = -x_log_x(1-alpha)
s_mix = s_mix_co2 + s_mix_other
s_mix_specific_co2 = s_mix.copy()
s_mix_specific_co2[1:] /= alpha[1:]
s_mix_co2_units = np.zeros_like(alpha)
s_mix_co2_units[1:-1] = -2494 * (np.log(alpha[1:-1]) + ((1-alpha[1:-1])/alpha[1:-1])*np.log(1-alpha[1:-1]))# Create figure and axes manually
fig = plt.figure(figsize=(6, 3))
ax = fig.add_axes([0.1, 0.2, 0.5, 0.7]) # [left, bottom, width, height]
# Plot the result
ax.plot(
alpha,
s_mix,
color='r',
linewidth=2,
)
# Set axis labels with LaTeX formatting
ax.set_xlabel(
r'Fraction of CO$_2$ $\alpha = n_{\text{CO2}} / n_{\text{total}}$',
fontsize=14,
)
ax.set_ylabel(
r'$\Delta S_{\text{mix}} / (n_{\text{total}} R)$',
fontsize=14,
labelpad=50,
rotation=0,
)
# Remove y-axis ticks except for y = 0
ax.set_yticks(np.atleast_1d(0))
# Show plot
plt.show()
# Create figure and axes manually
fig = plt.figure(figsize=(6, 5))
ax = fig.add_axes([0.1, 0.2, 0.8, 0.6]) # [left, bottom, width, height]
# Plot the contributions
ax.plot(
alpha,
s_mix_co2,
color=(0, 0.7, 1.0), # RGB color for CO2 contribution
linewidth=2,
label=r'CO$_2$ contribution ($s_\text{mix,CO2}$)',
)
ax.plot(
alpha,
s_mix_other,
color=(0, 0.7, 0), # RGB color for other gas contribution
linewidth=2,
label=r'Other gas contribution ($s_\text{mix,other}$)',
)
# Plot the total entropy of mixing
ax.plot(
alpha,
s_mix,
color='r',
linewidth=2,
label=r'Total entropy of mixing ($\Delta S_\text{mix}$)',
)
# Set axis labels with LaTeX formatting
ax.set_xlabel(
r'Fraction of CO$_2$ $\alpha = n_{\text{CO2}} / n_{\text{total}}$',
fontsize=14,
)
ax.set_ylabel(
r'$\Delta S_{\text{mix}} / (n_{\text{total}} R)$',
fontsize=14,
labelpad=50,
rotation=0,
)
# Remove y-axis ticks except for y = 0
ax.set_yticks(np.atleast_1d(0))
# Add legend
ax.legend(fontsize=12, loc='center left', bbox_to_anchor=(1.05, 0.5))
# Show plot
plt.show()
# Create figure and axes manually
fig = plt.figure(figsize=(6, 3))
ax = fig.add_axes([0.1, 0.2, 0.5, 0.7]) # [left, bottom, width, height]
# Plot the result
ax.plot(
alpha[1:],
s_mix_specific_co2[1:],
color='r',
linewidth=2,
)
# Set axis labels with LaTeX formatting
ax.set_xlabel(
r'Fraction of CO$_2$ $\alpha = n_{\text{CO2}} / n_{\text{total}}$',
fontsize=14,
)
ax.set_ylabel(
r'$\Delta \overline{S}_{\text{mix,CO2}} / R$',
fontsize=14,
labelpad=50,
rotation=0,
)
# Remove y-axis ticks except for y = 0
ax.set_yticks(np.atleast_1d(0))
# Show plot
plt.show()
# Create figure and axes manually
fig = plt.figure(figsize=(6, 3))
ax = fig.add_axes([0.1, 0.2, 0.5, 0.7]) # [left, bottom, width, height]
# Plot the result
ax.plot(
alpha[1:],
s_mix_co2_units[1:] / 1000,
color='r',
linewidth=2,
)
# Set axis labels with LaTeX formatting
ax.set_xlabel(
r'CO$_2$ Partial Pressure (atm)',
fontsize=14,
)
ax.set_ylabel(
r'$\Delta \overline{S}_{\text{mix,CO2}}$ (kJ/mol CO$_2$)',
fontsize=14,
labelpad=80,
rotation=0,
)
# Remove y-axis ticks except for y = 0
# ax.set_yticks(np.atleast_1d(0))
# Show plot
plt.show()
