## Integration of the heat capacity

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From thermodynamics, the heat capacity is defined as $$C_p = \left(\frac{dH}{dT}\right)_P$$. That means we can calculate the heat required to change the temperature of some material from the following integral:

$$H_2 - H_1 = Q = \int_{T_1}^{T_2} C_p(T) dT$$

In the range of 298-1200K, the heat capacity of CO2 is given by a Shomate polynomial:

$$C_p(t) = A + B t + C t^2 + D t^3 + E/t^2$$ with units of J/mol/K.

where $$t = T / 1000$$, and $$T$$ is the temperature in K. The constants in the equation are

value
A 24.99735
B 55.18696
C -33.69137
D 7.948387
E -0.136638
F -403.6075
G 228.2431
H -393.5224

## 1 Integrate the heat capacity

Use this information to compute the energy (Q in kJ/mol) required to raise the temperature of CO2 from 300K to 600K. You should use scipy.integrate.quad to perform the integration.

### 1.1 solution   solution

A =  24.99735
B =  55.18696
C = -33.69137
D =  7.948387
E = -0.136638
F = -403.6075
G =  228.2431
H = -393.5224

def Cp(T):
t = T / 1000
return A + B*t + C*t**2 + D*t**3 + E / t**2

from scipy.integrate import quad

dH, _ = quad(Cp, 300, 600)
print(f'The change in enthalpy is {dH / 1000:1.3f} kJ/mol')

The change in enthalpy is 12.841 kJ/mol



## 2 Verify via Δ H

The change in enthalpy (in kJ / mol) from standard state is

$$dH − dH_{298.15}= A t + B t^2/2 + C t^3/3 + D t^4/4 − E/t + F − H$$

again, $$t = T / 1000$$.

Use this equation to compute the change in enthalpy when you increase the temperature from 300 K to 600 K.

### 2.1 solution   solution

def dH(T):
t = T / 1000
return A * t + B*t**2 / 2 + C * t**3 / 3 + D * t**4 / 4 - E/t + F - H

print(f'The change in enthalpy is {dH(600) - dH(300):1.3f} kJ/mol')

The change in enthalpy is 12.841 kJ/mol



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