Problem B

The gas phase reaction A + B → C + D is examined in an ideally stirred batch reactor with a constant volume. At time t = 0 the reactor is rapidly filled with a mixture of equal amounts of A and B at a pressure of 100 kPa and a temperature of 25 °C. After one hour the concentration of A is measured as \(\frac{C_{A}}{C_{A0}}=0.15\). The reaction temperature is constant at 25 °C. The reaction follows the rate expression:

\(R = -r_{A}=-r_{B}=k{C_{A}}{C_{B}}\)

a) What is the relationship between \({C_{A}}\) and \({C_{B}}\)?

Since the initial concentrations of A and B are the same and the stoichiometry shows that when one mole of A is consumed so is one mole of B it follows that CA = CB. The common concentration of A and B will be called C in the following.

# Calculate initial concentration of A and B
def calculate_C0(C_ratio, n, P, R, T):
    C0 = (0.5*P)/(R*T)
    return round(C0, 2)

T1 = 25 # temperature in C
T1_kelvin = round(T1 + 273.15) # temperature in K

CA_CA0 = 0.15 # ratio of CA/CA0
R = 8.3144 # gas constant in Pa*m^3/(mol*K)
P = 100000 # pressure
C0 = calculate_C0(CA_CA0, 0.5, P, R, T1_kelvin)
print(f'The initial concentration of A and B is: {C0} mol/m^3')

The initial concentration of A and B is: 20.18 mol/m^3

b) Compute the rate constant k. The gas can be considered ideal. The gas constant is \(R = 8.3144 Pa\cdot m^3/(mol\cdot K)\)

def rate_constant(C0, C_ratio, t):
    k_h = (1/(C0*t))* (1/C_ratio -1)
    k_s = k_h/3600
    return k_s

t = 1 # time in h
k = rate_constant(C0, CA_CA0, t)
print(f'The constant rate k is: {k} m^3/mol*s')

The constant rate k is: 7.800168850713945e-05 m^3/mol*s

The activation energy \(E_a\) of the reaction is given by: \(E_a/R=10200 K\)

c) Calculate the rate constant at 100 °C

The gas constant is \(R = 8.3144 J/mol/K \)

import math

def rate_constant_100C(k, Ea_R, T1_kelvin, T2_kelvin):
    k_100 = k*math.exp((Ea_R*(1/T1_kelvin-1/T2_kelvin)))
    return k_100

Ea_R = 10200 # activation energy in K
T2 = 100 # temperature in C
T2_kelvin = round(T2 + 273.15) # temperature in K

print(f'The rate constant at 100 C is: {rate_constant_100C(k, Ea_R, T1_kelvin, T2_kelvin)} m^3/mol*s')

The rate constant at 100 C is: 0.07604449355095103 m^3/mol*s