6. ⇒ (MHT CET 2023 11th May Morning Shift )

A monoatomic gas at pressure ' $P$ ', having volume ' $V$ ' expands isothermally to a volume ' $2 V$ ' and then adiabatically to a volume ' $16 V$ '. The final pressure of the gas is (Take $γ = 5 / 3$ )

A. $P / 64$

B. $P / 32$

C. $16 P$

D. $32 P$

Correct Option is (A)

After isothermal expansion:

$\begin{aligned} P_{1} V_{1} = P_{2} V_{2} \\ P_{2} = P_{1} \frac{V_{1}}{V_{2}} \\ P_{2} = P_{1} \frac{V}{2 V} \\ P_{2} = \frac{P}{2} \end{aligned}$

After adiabatic expansion:

$\begin{aligned} P_{2} V_{2}^{γ} = P_{3} V_{3}^{γ} \\ P_{3} = P_{2} {(\frac{V_{2}}{V_{3}})}^{γ} \\ P_{3} = \frac{P}{2} {(\frac{2 V}{16 V})}^{5 / 3} \\ P_{3} = \frac{P}{2} {(\frac{1}{8})}^{5 / 3} \\ P_{3} = \frac{P}{64} \end{aligned}$

7. ⇒ (MHT CET 2023 11th May Morning Shift )

A diatomic gas $(γ = \frac{7}{5})$ is compressed adiabatically to volume $\frac{V_{i}}{32}$ where $V_{i}$ is its initial volume. The initial temperature of the gas is $T_{i}$ in Kelvin and the final temperature is ' $x T_{i}$ '. The value of ' $x$ ' is

A. 5

B. 4

C. 3

D. 2

Correct Option is (B)

For adiabatic process: ${TV}^{γ - 1} =$ Constant

$∴$ Initially:

$T_{i} V^{\frac{7}{5} - 1} = Constant$

$∴$ Final condition:

${xT}_{f} V^{\frac{7}{5} - 1} = Constant$

So,

$\begin{aligned} T_{i} V^{\frac{7}{5} - 1} = x T_{f} V^{\frac{7}{5} - 1} \\ T^{\frac{2}{5}} = x T {(\frac{V}{32})}^{\frac{2}{5}} \\ x = 4 \end{aligned}$

8. ⇒ (MHT CET 2023 10th May Evening Shift )

Which of the following graphs between pressure (P) and volume (V) correctly shows isochoric changes?

MHT CET 2023 10th May Evening Shift Physics - Heat and Thermodynamics Question 27 English

A. D

B. B

C. C

D. A

Correct Option is (C)

Isochoric processes are those in which the volume of the system remains constant. In a Pressure-Volume (P-V) graph, an isochoric process is represented by a vertical line because the volume does not change, while the pressure can vary.

Looking at the provided graph :

Graph A shows a diagonal line, indicating changes in both pressure and volume.

Graph B shows a horizontal line, indicating constant pressure, not constant volume. This represents an isobaric process.

Graph C shows a vertical line, which is characteristic of an isochoric process, where the volume remains constant while the pressure changes.

Graph D shows a curve, indicating that both pressure and volume are changing.

Therefore, the correct option that represents an isochoric process is :

Option C : C

9. ⇒ (MHT CET 2023 10th May Morning Shift )

A sample of gas at temperature $T$ is adiabatically expanded to double its volume. The work done by the gas in the process is $(\frac{C_{P}}{C_{V}} = γ = \frac{3}{2}) (R =$ gas constant $)$

A. $TR (\sqrt{2} - 2)$

B. $\frac{T}{R} (\sqrt{2} - 2)$

C. $\frac{R}{T} (2 - \sqrt{2})$

D. $RT (2 - \sqrt{2})$

Correct Option is (D)

Using the formula for adiabatic expansion,

$\begin{aligned} T_{1} V_{1}^{γ - 1} = T_{2} V_{2}^{γ - 1} \\ T_{1} V_{1}^{(1 / 2)} = T_{2} (2 V_{1}^{1 / 2}) . . . . . (Given : γ = \frac{3}{2}) \\ ∴ & T_{1} = T_{2} (\sqrt{2}) \\ ∴ & T_{2} = \frac{T}{\sqrt{2}} \end{aligned}$

Work done

$\begin{aligned} W_{adi} = & \frac{R (T - T_{2})}{γ - 1} = \frac{R (T - \frac{T}{\sqrt{2}})}{\frac{1}{2}} \\ = \frac{R (\sqrt{2} T - T)}{\sqrt{2}} \times 2 = RT (\sqrt{2} - 1) \sqrt{2} \\ ∴ W_{adi} & = RT (2 - \sqrt{2}) \end{aligned}$

10. ⇒ (MHT CET 2023 9th May Evening Shift )

A gas at N.T.P. is suddenly compressed to onefourth of its original volume. If $γ = 1.5$ , then the final pressure is

A. 4 times

B. 1.5 times

C. 8 times

D. $\frac{1}{4}$ times

Correct Option is (C)

As the change is sudden, the process is adiabatic

$\begin{array}{ll} ∴ & P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ} \\ ∴ & \frac{P_{2}}{P_{1}} = {[\frac{V_{1}}{V_{2}}]}^{γ} = {[\frac{4}{1}]}^{3 / 2} = \frac{8}{1} \end{array}$

⇒Thermodynamics