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JEE Advance - Chemistry (2023 - Paper 2 Online - No. 14)

The entropy versus temperature plot for phases $\alpha$ and $\beta$ at 1 bar pressure is given. $S_{\mathrm{T}}$ and $S_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{~K}$, respectively.


The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{~K}$ and $C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}=1 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. Assume $\left(C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}\right)$ is independent of temperature in the range of 200 to $700 \mathrm{~K} . C_{\mathrm{p}, \alpha}$ and $C_{\mathrm{p}, \beta}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.
The entropy versus temperature plot for phases $\alpha$ and $\beta$ at 1 bar pressure is given. $S_{\mathrm{T}}$ and $S_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{~K}$, respectively.


The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{~K}$ and $C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}=1 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. Assume $\left(C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}\right)$ is independent of temperature in the range of 200 to $700 \mathrm{~K} . C_{\mathrm{p}, \alpha}$ and $C_{\mathrm{p}, \beta}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.
The entropy versus temperature plot for phases $\alpha$ and $\beta$ at 1 bar pressure is given. $S_{\mathrm{T}}$ and $S_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{~K}$, respectively.


The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{~K}$ and $C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}=1 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. Assume $\left(C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}\right)$ is independent of temperature in the range of 200 to $700 \mathrm{~K} . C_{\mathrm{p}, \alpha}$ and $C_{\mathrm{p}, \beta}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.
The value of entropy change, $S_\beta-S_\alpha$ (in $\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$ ), at $300 \mathrm{~K}$ is _______.

[Use : $\ln 2=0.69$

Given : $S_\beta-S_\alpha=0$ at $0 \mathrm{~K}$ ]
Answer
0.31

Explanation

$\begin{aligned} & \text { At } 1 \text { bar } \\\\ & \alpha \longrightarrow \beta\end{aligned}$

$$ \begin{aligned} & \mathrm{S}_{\alpha(600)}^{\mathrm{o}}=\mathrm{S}_{\alpha(300)}^{\mathrm{o}}+\mathrm{C}_{\mathrm{P}(\alpha)} \ln \frac{600}{300} \\\\ & \mathrm{~S}_{\beta(600)}^{\mathrm{o}}=\mathrm{S}_{\beta(300)}^{\mathrm{o}}+\mathrm{C}_{\mathrm{P}(\beta)} \ell \mathrm{n} \frac{600}{300} \\\\ & \mathrm{~S}_{\beta(600)}^{\mathrm{o}}-\mathrm{S}_{\alpha(600)}^{\mathrm{o}}=\mathrm{S}_{\beta(300)}^{\mathrm{o}}-\mathrm{S}_{\alpha(300)}^{\mathrm{o}}+\left(\mathrm{C}_{\mathrm{P}(\beta)}-\mathrm{C}_{\mathrm{P}(\alpha)}\right) \ell \mathrm{n} 2 \\\\ & 6-5=\mathrm{S}_{\beta(300)}^{\mathrm{o}}-\mathrm{S}_{\alpha(300)}^{\mathrm{o}}+1 \times \ell \mathrm{n} 2 \\\\ & 1=\mathrm{S}_{\beta(300)}^{\mathrm{o}}-\mathrm{S}_{\alpha(300)}^{\mathrm{o}}+0.69 \end{aligned} $$

So $\mathrm{S}_{\beta(300)}^{\mathrm{o}}-\mathrm{S}_{\alpha(300)}^{\mathrm{o}}=0.31$

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