First Law of Thermodynamics and Its Applications (Only for Ideal Gases)

One mole of an ideal gas undergoes two different cyclic processes $\mathrm{I}$ and $\mathrm{II}$, as shown in the $P-V$ diagrams below. In cycle $\mathrm{I}$, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle $\mathrm{II}$, processes $a\text{'}, b\text{'}, c\text{'}$ and $d\text{'}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle $\mathrm{I}$ is $W_{\mathrm{I}}$ and that during cycle $\mathrm{II}$ is $W_{\mathrm{II}}$. The ratio $\frac{W_{\mathrm{I}}}{W_{\mathrm{II}}}$ is _____.

A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas $\left(\gamma =\frac{5}{3}\right)$ and one mole of an ideal diatomic gas $\left(\gamma =\frac{7}{5}\right)$. Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of $66$ Joule when heated at constant pressure. The change in its internal energy is _____ Joule.

One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $\frac{T_A}{T_B}$?

What is $∆U$ for the process described by figure. Heat supplied during the process is $q= 200 \mathrm{kJ}$.

How the first law of thermodynamics can be justified? Give an example in support of your answer.

In the given $P-V$ diagram, a monoatomic gas $\left(\gamma =\frac{5}{3}\right)$ is first compressed adiabatically from state $A$ state $B$. Then it expands isothermally from state $B$ to state $C$. [Given : ${\left(\frac{1}{3}\right)}^{0.6}=0.5,\ln 2\approx 0.7$].

Which of the following statement(s) is(are) correct?

An ideal gas of density $\rho =0.2 \mathrm{kg} {\mathrm{m}}^{-3}$ enters a chimney of height $h$ at the rate of $\alpha =0.8 \mathrm{kg} {\mathrm{s}}^{-1}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_1=0.1 {\mathrm{m}}^2$ and the upper end is $A_2=0.4 {\mathrm{m}}^2$. The pressure and the temperature of the gas at the lower end are $600 \mathrm{Pa}$ and $300 \mathrm{K}$, respectively, while its temperature at the upper end is $150 \mathrm{K}$. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and the ratio of specific heats of the gas $\gamma =2$. Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?

An idealized diesel engine operates in a cycle known as air standard diesel cycle as shown. Fuel is sprayed into the cylinder at the point of maximum compression $\mathrm{B}$. Combustion occurs during expansion $\mathrm{BC}$.

$n$ moles of an ideal gas is taken through a cyclic process $ABCD$. Pressure - density diagram of which is shown in adjacent figure. If molar mass of gas is $M$, what will be the work done by gas in cycle.

Figure shows a long fixed container which has two freely movable (without friction) pistons. The container and pistons are made up of a thermally conducting material, that allows very slow transfer of heat. First compartment of container is filled with $2$ moles of an ideal monoatomic gas at $200 \mathrm{K}$ and the $2^{\mathrm{nd}}$ compartment is filled with 1 moles of ideal diatomic gas at $500 \mathrm{K}$. Initially pressure of gases in both the compartment is same and equal to atmospheric pressure. Temperature of atmosphere is $300 \mathrm{K}$. Finally gases achieve equilibrium. '$R$' is the gas constant

A cylindrical container contains hydrogen in gaseous state under a light piston. The container and the piston both are very good insulators of heat. Initially the piston stays at a height $h_i$ above the bottom of the container in equilibrium. A ball of mass $m$ is dropped from a height $H$ above the piston collides elastically with the piston. When all motion ceases and equilibrium is reestablished, the piston again stays at the height $h_i.$
Neglect air resistance and take acceleration due to gravity as $g.$ Mark the CORRECT statement(s)

Two identical adiabatic vessels are filled with oxygen at pressure $P_1$ and $P_2\left(P_1>P_2\right).$ The vessels are interconnected with each other by a non-conducting pipe. If $U_{i_1}$ and $U_{i_2}$ denote initial internal energies of oxygen in first and second vessels respectively and $U_{f_1}$ and $U_{f_2}$ denote final internal energies respectively, then

A heat engine used one mole of an ideal monatomic gas as a working substance. The engine can follow either cycle $A$ or cycle $B$ in the diagram. The ratio of the efficiencies of the cycles is $\left(\frac{{\mathrm{e}}_{\mathrm{A}}}{{\mathrm{e}}_{\mathrm{B}}}\right)=4.$ If the value of $e_{\mathit{A}}+e_{\mathit{B}}$ is $\frac{n}{4}.$ Find the value of $n.$

An ideal gas can expand from state $A$ to $C$ (via state $B)$ through two different processes $P$ and $Q.$ In process $P, AB$ and $BC$ are straight lines in $P-V$ diagram. In process $Q,$ states

$A, B, C$ are on a circle in $P-V$ diagram. Heat supplied and work done by gas in process $P$ and $Q$ are respectively $Q_{\mathit{P}}, Q_{\mathit{Q}}$ and $W_P, W_Q \mathit{:}\mathit{-}$

One mole of a monoatomic ideal gas follows a process $A\to B,$ as shown $\left(A\equiv V_0,3P_0\right.$ and $\left.B\equiv 5V_0,n{P}_0\right).$ The average molar heat capacity for the process is $\frac{13R}{6}.$ The value of $\text{'}n\text{'}$ on $P-$axis (see following figure) is

A cylindrical vessel is divided in two parts by a fixed partition which is perfectly heat conducting. The wall and piston are thermally insulated from surroundings. The left side contains $0.5$ moles of gas with $C_v=2R$ at temperature of $300 \mathrm{K}.$ The right side contains $4$ moles of a mixture of gases with $C_v=1.75R$ at same temperature of $300 K.$ The piston compresses slowly the right side from volume of $V_0$ to $\frac{V_0}{4}.$ Find the total change in internal energy of gases. (take $R=\frac{25}{3} S.I.$ unit and consider ideal gases only)

A container filled with air under pressure $P_0$ contains a soap bubble of radius $R.$ The air pressure has been reduced to half isothermally and the new radius of the bubble becomes $\frac{5R}{4}.$ If the surface tension of the soap water solution is $S, P_0$ is found to be $\frac{16nS}{R}SI$ unit. Find $n.$

$P-V$ diagram of cyclic process $ABCA$ is as shown in figure. Choose the correct statement(s)

A large vacuum chamber of volume '$V$' is connected to a narrow cylindrical pipe which is initially filled with an ideal gas $\left(\gamma =1.4\right)$ at temperature $T_{0}$, pressure $P_0$ and volume $V_0\left(<<V\right)$. The gas is enclosed by a light insulating piston (which is free to move) and the air outside is constant pressure$P_0$. Now the gas in the pipe slowly allowed to leak into the vacuum chamber through a small opening as shown. All the walls are insulated. The final pressure '$P$' and temperature '$T$' of the gas are given by

Consider a thermodynamic cycle in a $PV$ diagram shown in the figure performe by one mole of a monoatomic gas. The temperature at $A$ is $T_0$ and volume at $A$ and $B$ are related as $V_B=V_C=2V_A$. Choose the correct option(s) form the following