Question

For a perfectly crytalline solid $C_{p,m}=a{T}^3,$ where $a$ is constant. If $C_{p,m}$ is $0.42J/Kmol$ at $10K_p$ molar entropy at $20K$ is:

• A. $0.42J/Kmol$
• B. $0.14J/Kmol$
• C. $1.12J/Kmol$
• D. zero

Answer 1

The correct option is C $1.12J/Kmol$

Solution:- (C) $1.12J/mol-K$

$C_{p,m}=a{T}^3$

Given that:-

$C_{p,m}$ at $10K=0.42J/mol-K$

$\therefore a=\frac{0.42}{{\left(10\right)}^3}=0.42\times {10}^{-3}$

As we know that,

$S_m={\int }_0^{20}\frac{C_{p,m}}{T}dT$

$S_m={\int }_0^{20}a{T}^{2}dT$

$S_m=a{\int }_0^{20}{T}^{2}dT$

$S_m=a\times {\left[\frac{T^3}{3}\right]}_0^{20}$

$\Rightarrow S_m=\frac{0.42\times {20}^{-3}}{3}({20}^3-0)=1.12J/K-mol$

Hence the molar entropy at $20K$ is $1.12J/mol-K$.