Question

The sample of monoatomic gas undergoes a process as represented by the $P–V$graph (if $P_0{V}_0=\frac{1}{3}R{T}_0$) then:

(P) $W_{1->2}=\frac{1}{3}R{T}_0$ (Q) $Q_{1->2->3}=\frac{11}{6}R{T}_0$ (R) $U_{1->2}=\frac{R{T}_0}{2}$(S) $W_{1->2->3}=\frac{1}{3}R{T}_0$

• A. P, Q, R, S are correct
• B. Only P, Q are correct
• C. Only R, S are correct
• D. Only P, R, S correct

Answer 1

The correct option is A

P, Q, R, S are correct

Step 1. Given Data,

$W_{1->2}=\frac{1}{3}R{T}_0$

$Q_{1->2->3}=\frac{11}{6}R{T}_0$

$U_{1->2}=\frac{R{T}_0}{2}$

$W_{1->2->3}=\frac{1}{3}R{T}_0$

Step 2. Formula used,

Work done by the gas, $W=PV$

$P$ is the pressure

$V$ is the volume

$Q$ is the heat given to the gas,

$Q=\mathrm{nC\Delta T}$

$n$ is the number of moles

$\mathrm{\Delta T}$ is the rise in temperature

$C$is the specific heat capacity

Step 3: Process: P Work done from process 1 to 2

$W_{1\to 2}=P_0{V}_0=\frac{1}{3}R{T}_0$

Step 4: Step 5: Process: Q

Heat change from process 1 to 2

${\mathrm{Q}}_{1\to 2}={\mathrm{nC}}_{\mathrm{p}}\mathrm{\Delta T}=\frac{5}{6}{\mathrm{RT}}_{\mathrm{o}}$

Heat change from process 2 to 3

${\mathrm{Q}}_{2\to 3}={\mathrm{nC}}_{\mathrm{v}}\mathrm{\Delta T}=\frac{\mathrm{f}}{2}\times 2{\mathrm{P}}_{\mathrm{o}}{\mathrm{V}}_{\mathrm{o}}={\mathrm{RT}}_{\mathrm{o}}$

Total heat change from process 1 to 2 to 3

${\mathrm{Q}}_{1\to 2\to 3}=\frac{5}{6}{\mathrm{RT}}_{\mathrm{o}}+{\mathrm{RT}}_{\mathrm{o}}=\frac{11}{6}{\mathrm{RT}}_{\mathrm{o}}$

Step 5 Process: R Internal energy change from process 1 to 2

$$\begin{gathered} \mathrm{\Delta U}={\mathrm{nC}}_{\mathrm{v}}\mathrm{\Delta T}=\mathrm{n}\frac{3}{2}{\mathrm{RT}}_{\mathrm{o}} \\ \mathrm{\Delta U}=\frac{{\mathrm{RT}}_{\mathrm{o}}}{2} \end{gathered}$$

Step 6 Process: S Work done from process 1 to 2 to 3

Work done from process 1 to 2 is given by:

$W_{1\to 2}=P_0{V}_0=\frac{1}{3}R{T}_0$

Work done from process 2 to 3 is zero as this process is isochoric i.e., the change in volume is zero.

$W_{1\to 2\to 3}=P_0{V}_0=\frac{1}{3}R{T}_0+0=\frac{1}{3}R{T}_0$

Therefore, the conditions for the processes P, Q, R, and S provided in the question are correct.

Therefore, the correct option is (A).