Solutions
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1. Define your own functions (with docstrings) to calculate the following quantities.
a. The de Broglie wavelength
\[\lambda = \frac{h}{p},\]
where \(h\) is Planck’s constant and \(p\) is the momentum.
Test your function for \(p = 1.01 \times 10^{-31}\,\mathrm{kg\ m\ s}^{-1}\).
from scipy import constants
def de_broglie_wavelength(p):
'''
Calculates de Broglie wavelength for a given momentum.
Arguments
---------
p: float
Momentum in kg m s^-1
Returns
-------
float
de Broglie wavelength in m
'''
return constants.h / p
momentum = 1.01E-31 # kg m s^-1
wavelength = de_broglie_wavelength(momentum)
print(f'For a particle with momentum {momentum} kg m s⁻¹')
print(f'The de Broglie wavelength is {wavelength:.3g} m')For a particle with momentum 1.01e-31 kg m s⁻¹
The de Broglie wavelength is 0.00656 mb. The change in the Gibbs free energy
\[\Delta G = -RT\ln K,\]
where \(R\) is the gas constant, \(T\) is the temperature and \(K\) is the equilibrium constant.
Test your function for \(T = 500\, \mathrm{K}\) and \(K = 1.15\).
import scipy.constants as constants
import math
def eq_to_gibbs(T, K):
'''
Calculates the change in Gibbs free energy associated
with a given equilibrium constant at a given temperature.
DG = -RTln(K)
Arguments
---------
T: float
Temperature in K
K: float
Equilibrium constant (dimensionless)
Returns
-------
float
Change in Gibbs free energy in kJ mol^-1
'''
# Delta G in J mol^-1
delta_G = - constants.R * T * math.log(K)
# Convert to kJ mol^-1
delta_G *= 1E-3
return delta_G
T = 500 # K
K = 1.15
delta_G = eq_to_gibbs(T, K)
print(f'At T={T:.3g} K, an equilibrium constant of {K:.3g} corresponds to')
print(f'a change in Gibbs free energy of {delta_G:.3g} kJ mol⁻¹')At T=500 K, an equilibrium constant of 1.15 corresponds to
a change in Gibbs free energy of -0.581 kJ mol⁻¹c. The mean activity according to the Debye-Huckel limiting law
\[\gamma_{\pm} = \exp\left[-|z_{+}z_{-}|A\sqrt{I}\right],\]
where \(z_{+}\) and \(z_{-}\) are the charges on the cations and anions respectively, \(I\) is the ionic strength and \(A\) is a constant that depends upon the solvent and the temperature.
Test your function with values of
\[ z_{+} = 1\ \ \ \ \ z_{-} = 2 \]
\[ A = 1.179\, \mathrm{M}^{\frac{1}{2}} \ \ \ \ \ I = 0.018\, \mathrm{M} \]
import math
def dh_mean_activity(zp, zm, A, I):
'''
Calculates the mean activity γ_± of a solution
using the Debye-Huckel limiting law.
Arguments
---------
zp: int
Charge of positive ion (dimensionless)
zm: int
Charge of negative ion (dimensionless)
A: float
Debye-Huckel constant for given solution
(solvent and temperature dependent) in M^{-1/2}
I: float
Ionic strength of solution in M
Returns
-------
float
Mean activity of solution (dimensionless)
'''
gamma_pm = math.exp(-abs(zp*zm) * A * math.sqrt(I))
return gamma_pm
# Charge on each ion - integers i.e. NOT in units
# of e
z_plus = 1
z_minus = 2
# Ionic strength in M
I = 0.018
# DH constant in M^{-1/2}
A = 1.179
gamma_pm = dh_mean_activity(z_plus, z_minus, A, I)
print(f'The mean activity is {gamma_pm:.2g}')The mean activity is 0.732a. The thermal de Broglie wavelength \(\Lambda\) describes the average de Broglie wavelength of particles in an idealgas at a given temperature:
\[\Lambda = \frac{h}{\sqrt{2\pi mk_{\mathrm{B}}T}},\]
where \(h\) is Planck’s constant, \(m\) is the mass of each particle, \(k_\mathrm{B}\) is Boltzmann’s constant and \(T\) is the temperature.
Write a function to calculate \(\Lambda\).
2b. The total entropy of an ideal gas can be calculated with the Sackur-Tetrode equation:
\[\frac{S}{k_{\mathrm{B}}N} = \ln\left(\frac{V}{N\Lambda^{3}}\right) + \frac{5}{2},\]
where \(S\) is the entropy, \(N\) is the number of particles, \(V\) is the volume, \(k_{\mathrm{B}}\) is Boltzmann’s constant and \(\Lambda\) is the thermal de Broglie wavelength.
Write a function to calculate the entropy of an ideal gas with the Sackur-Tetrode equation.
Using your two functions, calculate the entropy of an ideal gas with
import math
import scipy.constants as constants
def thermal_de_broglie_wavelength(m, T):
'''
Calculates the thermal de Broglie wavelength
of an ideal gas of given mass
at given temperature.
Arguments
---------
m: float
Mass of particle in kg
T: float
Temperature of system in K
Returns
-------
float
Thermal de Broglie wavelength in m
'''
denom = math.sqrt(2 * math.pi * m * constants.Boltzmann * T)
thermal_db = constants.h / denom
return thermal_db
def sackur_tetrode_entropy(V, thermal_db, N):
'''
Calculates the total entropy of an ideal gas
using the Sackur-Tetrode equation for a given
volume, number of particles, and thermal
de Broglie wavelength.
Arguments
---------
V: float
Volume of system in m^3
thermal_db: float
Thermal de Broglie wavelength in m
N: float | int
Number of particles
Returns
-------
float
Total entropy in J K^{-1}
'''
entropy = math.log(V / (N * thermal_db**3)) + 2.5
entropy *= constants.Boltzmann * N
return entropy
# System configuration
mass = 6.63E-26 # kg
T = 500 # K
N = 1.92E23
V = 5E-3
# Thermal de Broglie wavelength of gas
thermal_db = thermal_de_broglie_wavelength(mass, T)
# Entropy according to ST equation
entropy = sackur_tetrode_entropy(V, thermal_db, N)
print('For an ideal gas with the following attributes')
print(f' - Mass = {mass:.3g} kg')
print(f' - Temperature = {T:.3g} K')
print(f' - Number of Particles = {N:.3g}')
print(f' - Volume of container = {V:.3g} m³')
print(f'\nThe thermal de Broglie wavelength is {thermal_db:.3g} m')
print(f'and the Total (Sackur-Tetrode) Entropy is {entropy:.3g} J K⁻¹')For an ideal gas with the following attributes
- Mass = 6.63e-26 kg
- Temperature = 500 K
- Number of Particles = 1.92e+23
- Volume of container = 0.005 m³
The thermal de Broglie wavelength is 1.24e-11 m
and the Total (Sackur-Tetrode) Entropy is 50.2 J K⁻¹