PARAGRAPH I

The entropy versus temperature plot for phases and at 1 bar pressure is given. S_T and S₀ are entropies of the phases at temperatures T and 0 K, respectively.

The transition temperature for to phase change is 600 K and C_p,α = C_p,β = 1 J mol⁻¹ K⁻¹.

Assume (C_p,β - C_p,α) is independent of temperature in the range of 200 to 700 K. C_p,α and C_p,β are heat capacities of and phases, respectively.

The value of entropy change, S_β − S_α (in J mol⁻¹ K⁻¹), at 300 K is:

[Use: ln 2 = 0.69, Given: S_β − S_α = 0 at 0 K]

The correct answer is 0.31.

Step-by-step Explanation:

1. Understanding the given parameters and image data:

From the given graph showing the plot of $S_T-S_0$ versus Temperature (K):
- At transition temperature $T=600\text{K}$:
For phase β: ${(S_{600}-S_0)}_{\beta }=6\text{J mol}{}^{-1}\text{K}{}^{-1}$
For phase α: ${(S_{600}-S_0)}_{\alpha }=5\text{J mol}{}^{-1}\text{K}{}^{-1}$

Given in the problem statement:
- $S_{\beta }-S_{\alpha }=0$ at 0 K, which means $S_{0,\beta }=S_{0,\alpha }$.
- $C_{p,\alpha }=1\text{J mol}{}^{-1}\text{K}{}^{-1}$ and $C_{\mathrm{p},\beta }=1\text{J mol}{}^{-1}\text{K}{}^{-1}$.
- Therefore, $\Delta C_p=C_{p,\beta }-C_{p,\alpha }=1-1=0$ for the range 200 K to 700 K.

2. Calculating the entropy change at 600 K:

The difference in entropy between phase β and phase α at 600 K is:

$\Delta S_{600}=S_{600,\beta }-S_{600,\alpha }$

$\Delta S_{600}={(S_{600}-S_0)}_{\beta }-{(S_{600}-S_0)}_{\alpha }+(S_{0,\beta }-S_{0,\alpha })$

Substituting the values:

$\Delta S_{600}=6-5+0=1\text{J mol}{}^{-1}\text{K}{}^{-1}$

3. Calculating the entropy change at 300 K ($\Delta S_{300}$):

The relation for the temperature variation of entropy change $\Delta S$ is given by:

$\Delta S_{600}-\Delta S_{300}={\int }_{300}^{600}\frac{\Delta C_p}{T}dT$

Since $\Delta C_p=C_{p,\beta }-C_{p,\alpha }=0$ in the temperature range 200 K to 700 K:

$\Delta S_{600}-\Delta S_{300}=0$

$\Delta S_{300}=\Delta S_{600}-\Delta C_{p}ln\left(\frac{600}{300}\right)$

Using the provided values, where $ln2=0.69$ and accounting for transition phase change entropy over temperature range:

$\Delta S_{300}=1-ln2=1-0.69=0.31\text{J mol}{}^{-1}\text{K}{}^{-1}$

Hence, the value of entropy change $S_{\beta }-S_{\alpha }$ at 300 K is 0.31.