GATE 2026 PH – Question 33
The specific heat $C_p(T)$ of one mole of a material as a function of temperature $T$ is given as $C_p(T)=AT+BT^3$, where $A=0.695\ \mathrm{mJ\,mol^{-1}K^{-2}}$ and $B=0.045\ \mathrm{mJ\,mol^{-1}K^{-4}}$. When $T$ is changed from 1 K to 10 K at constant pressure, then the change in entropy $\Delta S$ in $\mathrm{mJ\,mol^{-1}K^{-1}}$ (rounded off to one decimal place) is _____
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Correct answer: 21.09 to 21.31
Explanation
At constant pressure the entropy change is
$$\Delta S=\int_{T_1}^{T_2}\frac{C_p}{T}\,dT .$$
**Integrate** $C_p=AT+BT^3$:
$$\Delta S=\int_1^{10}\left(A+BT^2\right)dT=A\,(10-1)+\frac B3\,(10^3-1^3).$$
**Numbers.** $A=0.695$ and $B=0.045$ (mJ mol⁻¹ K⁻² and K⁻⁴):
$$\Delta S=0.695\times9+\frac{0.045}{3}\times999=6.255+14.985=\mathbf{21.2}\ \text{mJ mol}^{-1}\text{K}^{-1}.$$