Embibe Experts Solutions for Chapter: Kinetic Theory of Gases and Thermodynamics, Exercise 2: Exercise 2

The heat capacity of one mole of an ideal gas is found to be $C_{\mathrm{V}}=\frac{3R\left(1+aRT\right)}{2}$ where $a$ is a constant. The equation obeyed by this gas during a reversible adiabatic expansion is-

A long cylindrical pipe of radius $20 \mathrm{cm}$ is closed at its upper end and has an airtight piston of negligible mass as shown. When a $50 \mathrm{kg}$ mass is attached to the other end of the piston, it moves down. If the air in the enclosure is cooled from temperature $T$ to $T-\mathrm{\Delta }T$, the piston moves back to its original position. Then $\frac{\mathrm{\Delta }T}{T}$ is close to (Assuming air to be an ideal gas, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$, atmospheric pressure is ${10}^5$ $\mathrm{Pascal}$),

Two different liquids of same mass are kept in two identical vessel, which are placed in a freezer that extracts heat from them at the same rate causing each liquid to transform into a solid. The schematic figure below shows the temperature $T$ vs time $t$ plot for the two materials. We denote the specific heat in the liquid states to be $C_{\mathrm{L}1}$ and $C_{\mathrm{L}2}$ for materials $1$ and $2$ respectively, and latent heats of fusion $U_1$ and $U_2$ respectively. Choose the correct option.

A thin piece of thermal conductor of constant thermal conductivity insulated on the lateral sides connects two reservoirs which are maintained at temperatures $T_1$ and $T_2$ as shown. Assuming that the system is in steady state, which of the following plots best represents the dependence of the rate of change of entropy on the ratio of temperatures $\frac{T_1}{T_2}$

Two bottles $A$ and $B$ have radii $R_A$ and $R_B$ and heights $h_A$ and $h_B$ respectively with $R_B=2R_A$ and $h_B=2h_A$. These are filled with hot water at $60°\mathrm{C}$. Consider that heat loss for the bottles takes place only from side surfaces. If the time the water takes to cool down to $50°\mathrm{C}$ is $t_A$ and $t_B$ for bottles $A$ and $B$, respectively, then $t_A$ and $t_B$ are best related as,

The number of gas molecules striking per second per square meter of the top surface of a table placed in a room at $20°\mathrm{C}$ and $1$ atmospheric pressure is of the order of ($k_{\mathrm{b}}=1.4\times {10}^{-23} \mathrm{J} {\mathrm{K}}^{-1}$ and the average mass of an air molecule is $5\times {10}^{-27} \mathrm{kg}$) 

One mole of an ideal monatomic gas undergoes the following four reversible processes:

Step 1: It is first compressed adiabatically from volume $V_1$ to $1 {\mathrm{m}}^3$.

Step 2 : then expanded isothermally to volume $10 {\mathrm{m}}^3$

Step 3 : then expanded adiabatically to volume $V_4$

Step 4 : then compressed isothermally to volume $V_1$.

If the efficiency of the above cycle is $\frac{3}{4}$ then $V_1$ is

A gas obeying the equation of state $PV=RT$ undergoes a hypothetical reversible process described by the equation, $P{V}^{\frac{5}{3}}\exp \left(-\frac{PV}{E_0}\right)=C_1$, where $C_1$ and $E_0$ are dimensioned constants. Then, for this process, the thermal compressibility at high temperature