At 300K, entropy of two allotropic form of a substance X and Y are 32.04 and 32.68 J/mol-K respectively. Heat of combustion are – 297.9 and –298.3 kJ/mol respectively. Then \DeltaG for the change X

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Calculation of Various Parameters at Equilibrium

Gibbs Free Energy Calculations and Third Law

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Question

At 300K, entropy of two allotropic form of a substance X and Y are 32.04 and 32.68 J/mol-K respectively. Heat of combustion are – 297.9 and –298.3 kJ/mol respectively. Then $∆$ G for the change
X —→ Y will be -

592 J/mol

0.208 J/mol

208 J/mol

208 kJ/mol

For allotropic change X —→ Y

$∆$ S = S_Y – S_X = 32.68 = 32.04

= 0.64 J/mol-K

$∆$ H = $∆$ _CH(X) – $∆$ _CH(Y)

= – 297.9 + 298.3

$∆$ H = 0.4 KJ/mol

So, $∆$ G = $∆$ H – T $∆$ S

$= 0 .4 - \frac{300 \times 0 .64}{1000} = 0 .208 kJ / mol$

This question involves calculating the Gibbs free energy change (ΔG) for the transformation between two allotropic forms of a substance. The key concept is that the transformation X → Y can be considered as part of a thermodynamic cycle involving combustion. Since both allotropes combust to the same products, the difference in their combustion energies equals the enthalpy change for the transformation X → Y.

Step 1: Find ΔH for the reaction X → Y
The heat of combustion for X is -297.9 kJ/mol and for Y is -298.3 kJ/mol. Since combustion is exothermic, a more negative value means more energy is released. Therefore, Y is more stable than X. The enthalpy change for X → Y is: $∆ H = ∆_{H} {(c o m b u s t i o n)}_{X} - ∆_{H} {(c o m b u s t i o n)}_{Y} = (- 297.9) - (- 298.3) = 0.4 k J / m o l$

Step 2: Find ΔS for the reaction X → Y
The entropy for X is 32.04 J/mol-K and for Y is 32.68 J/mol-K. The entropy change is: $∆ S = S_{Y} - S_{X} = 32.68 - 32.04 = 0.64 J / m o l - K$

Step 3: Apply the Gibbs Free Energy Equation
The temperature T is 300 K. The formula for Gibbs free energy is: $∆ G = ∆ H - T ∆ S$
Substitute the values. Remember to use consistent units. Convert ΔH from kJ to J (1 kJ = 1000 J). $∆ G = (0.4 \times 1000) - (300 \times 0.64) = 400 - 192 = 208 J / m o l$

Final Answer: The ΔG for the change X → Y is 208 J/mol.

Related Concepts and Formulae

Gibbs Free Energy (ΔG): This is a thermodynamic potential that predicts the spontaneity of a process at constant temperature and pressure. A negative ΔG indicates a spontaneous process.

Key Formula: $∆ G = ∆ H - T ∆ S$
Where:
ΔH = Enthalpy change (J/mol)
T = Absolute Temperature (K)
ΔS = Entropy change (J/mol-K)

Allotropes: Different structural forms of the same element in the same physical state (e.g., graphite and diamond for carbon). They have different thermodynamic properties.

Hess's Law: The total enthalpy change for a reaction is the same regardless of the number of steps taken. This principle was used indirectly by recognizing that the difference in combustion enthalpies equals the enthalpy of transformation.

Detailed Solution

Related Concepts and Formulae

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