Thermodynamics, Heat capacities and change in internal energy

Question

The Heat capacities at constant volume of gaseous A. B and C were shown below, calculate the value of ▲U for the reaction

A B —► C at 400K

given that, at 300K, ▲U= 85.6 kJ

For:

A Cv/JK^-1mol^-1= 15.1 6.4 X 10^-3T

B Cv/JK^-1mol^-1= 12.8 4.7 X 10^-3T

C Cv/JK^-1mol^-1= 14.3 2.0 X 10^-3T

Thanks in advance if anyone understands this

Answer 1

Assuming ideal gas internal energy is solely a function of the temperature:

dU/dT = Cv => ∫ dU = ∫ Cv dT

That's why you can calculate internal energy at any temeperature, if you know the internal energy at one temperature and the heta capacity Cv::

U(T₂) = U(T₁) + ∫ |T₁→T₂| Cv dT

Here we have a linear dependency of Cv on temperature:

Cv = α + β∙T

So

U(T₂) = U(T₁) + ∫ |T₁→T₂| α + β∙T dT

= U(T₀) + α∙(T₂ - T₁) + (1/2)∙β∙(T₂² - T₁²)

These relations hold for the internal energy of formation of the compound, which are considered for calculation of ΔU .You find the change of internal energy due to reaction by subtracting the internal energy of the reactants from those of the products. So the change of ΔU with temperature is given by the difference in the heat capacities:

ΔU(T₂) = U_C(T₂) - U_A(T₂) - U_B(T₂)

=  U_C(T₁) - U_A(T₁) - U_B(T₁) + (α_C - α_A - α_B)∙(T₂ - T₁) + (1/2)∙(β_C - β_A - β_B)∙(T₂² - T₁²)

= ΔU(T₁) + Δα∙(T₂ - T₁) + (1/2)∙Δβ∙(T₂² - T₁²)

with

Δα = 14.3JK⁻¹mol⁻¹ - 15.1JK⁻¹mol⁻¹ -  12.8JK⁻¹mol⁻¹

= -13.6JK⁻¹mol⁻¹

Δβ = 2.0×10⁻³JK⁻²mol⁻¹ - 6.4×10⁻³JK⁻²mol⁻¹ -  4.7×10⁻³JK⁻²mol⁻¹

= -9.1.0×10⁻³JK⁻²mol⁻¹

Therefore:

ΔU(400K) = ΔU(300K) - 0.0136kJK⁻¹mol⁻¹∙(400K - 300K) - (1/2)∙ 9.1.0×10⁻⁶kJK⁻²mol⁻¹∙((400K)² - (300k)²)

= 85.6kJmol⁻¹ - 1.997kJmol⁻¹

= 83.6kJmol⁻¹