Embibe Experts Solutions for Chapter: Thermodynamics, Exercise 3: Exercise - 3

The specific heat capacity of a metal at low temperature $\left(T\right)$ is given as $C_p\left(\mathrm{kJ} {\mathrm{K}}^{-1} {\mathrm{kg}}^{-1}\right)=32{\left(\frac{T}{400}\right)}^3$. A $100 \mathrm{g}$ vessel of this metal is to be cooled from $20 \mathrm{K}$ to $4 \mathrm{K}$ by a special refrigerator operating at room temperature $\left(27 °\mathrm{C}\right)$. The amount of work required to cool the vessel is

Find the coefficient of volume expansion of the gas, when an ideal gas is expanding such that $P{T}^2=$constant.

$C_v$ and $C_p$ denote the molar specific heat capacities of a gas at constant volume and constant pressure, respectively. Then

Find the work done in the process is, $5.6 \mathrm{L}$ of helium at $\mathrm{STP}$ is adiabatically compressed to $0.7 \mathrm{L}$. Taking the initial temperature to be $T_1$
 

Two moles of ideal helium gas are in a rubber balloon at $30°\mathrm{C}$. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to $35°\mathrm{C}$. The amount of heat required in raising the temperature is nearly (Take $R=8.31 \mathrm{J} {(\mathrm{mol}-\mathrm{K})}^{-1}$).
 

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $p_i={10}^5 \mathrm{Pa}$ and volume $V_i={10}^{-3} {\mathrm{m}}^3$ changes to a final state at $p_f=\left(\frac{1}{32}\right)\times {10}^5 \mathrm{Pa}$ and $V_f=8\times {10}^{-3} {\mathrm{m}}^3$ in an adiabatic quasi-static process, such that $p^3{V}^5=$constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $p_i$ followed by an isochoric (isovolumetric) process at volume $V_f$. The amount of heat supplied to the system in the two-step process is approximate,

Two rigid boxes containing different ideal gases are placed on a table. The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of boxes). Box A contains one mole of nitrogen at temperature $T_0$, while box $B$ contains one mole of helium at temperature $\frac{7}{3}{\mathit{ }\mathit{T}}_0$. Then the final temperature of the gases, $T_{\mathit{f}}$ in terms of $T_0$ is

A solid of constant heat capacity $1 \mathrm{J}{}^{\circ }\mathrm{C}^{-1}$ 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 the same amount of heat. In both the cases body is brought from initial temperature $100°\mathrm{C}$ to final $200°\mathrm{C}$. Entropy change of the body in the two cases respectively is :