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

In the given figure an ideal gas changes its state from $A$ to state $C$ by two paths $ABC$ and $AC$.

(i) The internal energy of gas at $A$ is $10 \mathrm{J}$ and amount of heat supplied to change its state to $C$ through the path $AC$ is $200 \mathrm{J}$. Calculate the internal energy at $C$.

(ii) The internal energy of gas at $B$ is $20 \mathrm{J}$. Find the amount of heat supplied to the gas from $A$ to $B$.

The pressure in monoatomic gas increases linearly from $4\times {10}^5 \mathrm{N} {\mathrm{m}}^{-2}$ to $8\times {10}^5 \mathrm{N} {\mathrm{m}}^{-2}$ when its volume increases from $0.2 {\mathrm{m}}^3$ to $0.5 {\mathrm{m}}^3$.

(i) If work done by the gas is $1.8\times {10}^{\mathrm{y}} \mathrm{J}$ then find the value of $y$.

(ii) If increase in internal energy is $1.6z\times {10}^5 \mathrm{J}$ then find the value of $z$.

(iii) If amount of heat supplied is $6.6\times {10}^{\mathrm{x}} \mathrm{J}$ then find the value of $x$.

(iv) If molar specific heat of the gas is $1.71\times {10}^{\mathrm{\alpha }} \mathrm{J} \mathrm{mole}-{\mathrm{K}}^{-1}$ then find the value of $\alpha$.

On mole of a monoatomic ideal gas is taken through the cycle shown in figure.

$A\to B$ : Adiabatic expansion            $B\to C$ : Cooling at constant volume 

$C\to D$ : Adiabatic compression

$D\to A$ : Heating at constant volume

The pressure and temperature at $A, B$, etc., are denoted by $P_A,T_A,P_B,T_B$ etc., respectively. Given that $T_A=1000 \mathrm{K}$, $P_B=\left(\frac{2}{3}\right)P_A$ and $P_C=\left(\frac{1}{3}\right)P_A$.

Calculate the following quantities : (i) The work done by the gas in the process $A\to B$

(ii) The heat lost by the gas in the process $B\to C$

(iii) The temperature $T_D$. (Given : ${\left(\frac{2}{3}\right)}^{2/5}=0.85$)

Three moles of an ideal gas $\left(C_P=\frac{7}{2}R\right)$ at pressure, $P_A$ and temperature $T_A$ is isothermally expanded to twice its initial volume. It is then compressed at constant pressure to its original volume. Finally gas is compressed at constant volume to its original pressure $P_A$. If the net work done by the gas, and net heat support to the gas during the complete process is $x$ and $y$ respectively then fine $\frac{x}{y}$.

Two moles of helium gas $\left(\gamma =5/3\right)$ are initially at temperature $27°\mathrm{C}$ and occupy a volume of $20 \mathrm{L}$. The gas is first expanded at constant pressure until the volume is doubled. Then it undergoes an adiabatic change until the temperature returns to its initial value. (i) If the final volume and pressure of the gas are $1.13\times {10}^{\alpha }\mathrm{L}$ and $4.4\times {10}^{\mathrm{\beta }} \mathrm{N} {\mathrm{m}}^{-2}$ then the find $\alpha +\beta$. (ii) What is the work done by the gas?

A gaseous mixture enclosed in a vessel of volume $V$ consists of one gram mole of gas. $A$ with $\gamma =\frac{C_P}{C_V}=\frac{5}{3}$ an another gas $B$ with $\gamma =\frac{7}{5}$ at a certain temperature $T$. The gram molecular weight of the gases $A$ and $B$ are $4$ and $32$ respectively. The gases $A$ and $B$ do not react with each other and are assumed to be ideal. The gaseous mixture follows the equation $P{V}^{19/13}=$ constant, in adiabatic process. Find the number of gram moles of the gas $B$ in the gaseous mixture.

One mole of an ideal gas is heated isobarically from the freezing point to the boiling point of water each under normal pressure. Find out the work done by the gas the change in its internal energy. The amount of heat involved is $1 \mathrm{kJ}$.

An ideal gas at $\mathrm{NTP}$ is enclosed in a adiabatic vertical cylinder having are of cross section $A=27 {\mathrm{cm}}^2$, between two light movable pistons as shown in the figure. Spring with force constant $k=3700 \mathrm{N} {\mathrm{m}}^{-1}$ is in a relaxed state initially. Now the lower piston is moved upwards a height $\frac{h}{2}$, being the initial length of gas column. It is observed that the upper piston moves up by a distance $\frac{h}{16}$. Find $h$ taking $\gamma$ for the gas to be $1.5.$ Also find the final temperature of the gas.