Question

A solid body of constant heat capacity $1 \, J/^{\circ}C$ is being heated by keeping it in contact with reservoirs in two ways (i) Sequentially keeping in contact with $2$ reservoirs such that each reservoir supplies same amount of heat. (ii) Sequentially keeping in contact with $8$ reservoirs such that each reservoir supplies same amount of heat. In both the cases, body is brought from initial temperature $100^{\circ}C$ to final temperature $200^{\circ}C$. Entropy change of the body in the two cases respectively, is

• A. In (2), In 2
• B. In 2, 2 ln 2
• C. 2 ln 2, 8 In 2
• D. In 2,4 In 2

Answer 1

Answer: (A)

In (2), In 2

Answer 2

(i) $\Delta S _{1}=\int \frac{ dQ }{ T }= ms \displaystyle\int_{100}^{150} \frac{ d T }{ T }+ ms \displaystyle\int_{150}^{200} \frac{ d T }{ T }$
$=\ln \left(\frac{150}{100}\right)+\ln \left(\frac{200}{150}\right)$
$=\ln \left(\frac{3}{2}\right)+\ln \frac{4}{3}$
$\Delta S _{1}=\ln 2$
(ii) $\Delta s _{2}=\int \frac{ dQ }{ T }=\displaystyle\int_{100}^{112.5} \frac{ dQ }{ T }+\displaystyle\int_{112.5}^{125} \frac{ dQ }{ T }+\ldots \ldots$
$=\ln \left(\frac{112.5}{100}\right)+\ln \left(\frac{125}{112.5}\right)+\ldots . .$
$=\ln \left(\frac{9}{8}\right)+\ln \left(\frac{10}{9}\right)+\ln \left(\frac{16}{15}\right)$
$=\ln \left(\frac{16}{8}\right)=\ln 2$