Question

Thermal decomposition of gaseous X₂ to gaseous X at 298 K takes place according to the following equation:

X₂ (g) ⇌ 2X (g)

The standard reaction Gibbs energy, Δ_rG°, of this reaction is positive. At the start of the reaction, there is one mole of X₂ and no X. As the reaction proceeds, the number of moles of X formed is given by β. Thus, β_equilibrium is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2 bar. Consider the gases to behave ideally.

[Given : R = 0.083 L bar K^–1 mol^–1]

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Linked Question 1

The equilibrium constant K_P for this reaction at 298 K, in terms of β_equilibrium, is

$\frac{4 β_{equilibrium}^{2}}{4 - β_{equilibrium}^{2}}$

$\frac{8 β_{equilibrium}^{2}}{2 - β_{equilibrium}}$

$\frac{8 β_{equilibrium}^{2}}{4 - β_{equilibrium}^{2}}$

$\frac{4 β_{equilibrium}^{2}}{2 - β_{equilibrium}^{}}$

$∆$ G° > 0

X₂ (g) $\underset{}{⇌}$ 2X (g) P = 2 bar

t = 0 1 mol

t = t $(\frac{1 - α}{1 + α}) \times 2$ $(\frac{2 α}{1 + α}) \times 2$

K_P = $\frac{{(\frac{2 α}{1 + α})}^{2} \times 4}{(\frac{1 - α}{1 + α}) \times 2} = \frac{8 α^{2}}{1 - α^{2}}$

2α = β_equilibrium

_{$K_{P} = \frac{8 {(\frac{β}{2})}^{2}}{1 - {(\frac{β}{2})}^{2}} \Rightarrow (\frac{8 β_{equilibrium}^{2}}{4 - β_{equilibrium}^{2}})$}

Linked Question 2

The INCORRECT statement among the following, for this reaction, is

At the start of the reaction, dissociation of gaseous X2 takes place spontaneously

β_equilibrium = 0.7

Decrease in the total pressure will result in formation of more moles of gaseous X

K_C < 1

For (A) option :

Q_P α P_total so on decreasing P_Total, Q_P decreases so reaction shift toward forward direction and form more moles of X(g)

For (B) option :

$∆$ G = $∆$ G° + RT ln Q_P

At the start, Q_P → 0

$∆$ G = $∆$ G° > 0, non spontaneous

For (C) option :

If β_equilibrium = 0.7

K_P = $[\frac{8 \times {(0 .7)}^{2}}{4 - {(0 .7)}^{2}}] \approx 1 .11$

K_P > 1 , C is incorrect

For (D) option :

$∆$ n_g = +1

$∆$ G° = –– RT ln K_P = +ve

ln K_P < 0 ⇒ K_P < 1

K_P = K_C (RT)^{$∆$ ng}

K_P = K_C (RT)

K_C = $\frac{K_{P}}{RT}$ < 1