Question

The following equation represents the decomposition of a generic diatomic element in its standard state.

1/2 X₂ (g) = X (g)

Assume that the standard molar Gibbs energy of formation of X(g) is 5.40 kJ·mol–1 at 2000. K and –53.08 kJ·mol–1 at 3000. K. Determine the value of K (the thermodynamic equilibrium constant) at each temperature.

K at 2000. K = ?

K at 3000. K = ?

Assuming that ΔH°rxn is independent of temperature, determine the value of ΔH°rxn from these data.

ΔH°rxn = ? kj x mol^-1

Answer 1

The following equation represents the decomposition of a generic diatomic element in its standard state.

1/2 X₂ (g) = X (g)

Assume that the standard molar Gibbs energy of formation of X(g) is 5.40 kJ·mol–1 at 2000. K and –53.08 kJ·mol–1 at 3000. K. Determine the value of K (the thermodynamic equilibrium constant) at each temperature.

Equilibrium constant and Standard free energy change of reaction are related as:
ΔrG = - R∙T∙ln(K)
<=>
K = e^{ - ΔrG/(R∙T) }

For this reaction at 2000 K , ΔrG = 5.40 kJ·mol–1

K = e^{ - (5.40×10³ J∙mol⁻¹ / ( 8.3145 J∙K⁻¹∙mol⁻¹ ∙ (2000)K ) }
K = e^-0.33

K= 0.72

For this reaction at 3000 K , ΔrG = –53.08 kJ·mol–1

K = e^{ - (-53.08×10³ J∙mol⁻¹ / ( 8.3145 J∙K⁻¹∙mol⁻¹ ∙ (3000)K ) }
K = e^+2.13

K= 8.42

Assuming that ΔH°rxn is independent of temperature, determine the value of ΔH°rxn from these data.

Use Van't Hoff equation:

ln(K₂/K₁) = - (∆H_rxn/R)∙((1/T₂) - (1/T₁))

ln(8.42/0.72) = - (∆H_rxn/8.3145 J∙K⁻¹∙mol⁻¹)∙((1/3000) - (1/200)

2.46= - (∆H_rxn/8.3145 J∙K⁻¹∙mol⁻¹)∙(3.3*10^-4-5*10^-4)

2.46= - (∆H_rxn/8.3145 J∙K⁻¹∙mol⁻¹)∙(-1.7*10^-4)

2.46/ 1.7*10^-4 K=(∆H_rxn/8.3145 J∙K⁻¹∙mol⁻¹)∙

∆H_rxn= 120315.7 J∙∙mol⁻¹ or 120.3 J∙mol⁻¹