In the given diagram, a monoatomic gas is first compressed adiabatically from state to state . Then it expands isothermally from state to state . Which of the following statement(s) is(are) correct? [2022]

Question

In the given $P - V$ diagram, a monoatomic gas $(γ = \frac{5}{3})$ is first compressed adiabatically from state $A$ to state $B$ . Then it expands isothermally from state $B$ to state $C$ .

$[Given {(\frac{1}{3})}^{0.6} = 0.5, \ln 2 = 0.7]$

Which of the following statement(s) is(are) correct? [2022]

Smart Achievers · Accepted Answer

(2, 3, 4)

$(1) W_{A \to B \to C} = W_{A \to B} + W_{B \to C}$

$= \frac{P_{A} V_{A} - P_{B} V_{B}}{γ - 1} + P_{A} V_{A} \ln (\frac{V_{C}}{V_{B}})$

$As, P_{A} V_{A}^{γ} = P_{B} V_{B}^{γ} \Rightarrow {(\frac{V_{B}}{V_{A}})}^{γ} = \frac{P_{A}}{P_{B}}$

$\Rightarrow V_{B} = {(\frac{P_{A}}{P_{B}})}^{1 / γ} V_{A} = 0.8 {(\frac{1}{3})}^{3 / 5} = 0.8 \times 0.5 = 0.4 m^{3}$

$So, W_{A \to B \to C} = \frac{100 \times 0.8 - 300 \times 0.4}{\frac{5}{3} - 1} + 300 \times 0.4 \ln (\frac{0.8}{0.4})$

$= \frac{- 40}{2 / 3} + 120 \ln 2 = - 60 + 120 \times 0.7 = - 60 + 84 = 24 kJ$

$So, (2,3,4) are correct.$

Solution

1 The magnitude of the total work done in the process $A \to B \to C$ is $144 k J$ .

2 The magnitude of the work done in the process $B \to C$ is $84 k J$ .

3 The magnitude of the work done in the process $A \to B$ is $60 k J$ .

4 The magnitude of the work done in the process $C \to A$ is zero.

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Solution

Q. In the given P-V diagram, a monoatomic gas (γ=53) is first compressed adiabatically from state A to state B. Then it expands isothermally from state B to state C. [Given (13)0.6=0.5, ln2=0.7] Which of the following statement(s) is(are) correct [2022]

1 The magnitude of the total work done in the process A→B→C is 144 kJ.

2 The magnitude of the work done in the process B→C is 84 kJ.

3 The magnitude of the work done in the process A→B is 60 kJ.

4 The magnitude of the work done in the process C→A is zero.

Ans. (2, 3, 4) (1) WA→B→C=WA→B+WB→C =PAVA-PBVBγ-1+PAVAln(VCVB) As, PAVAγ=PBVBγ⇒(VBVA)γ=PAPB ⇒VB=(PAPB)1/γVA=0.8(13)3/5=0.8×0.5=0.4 m3 So, WA→B→C=100×0.8-300×0.453-1+300×0.4ln(0.80.4) =-402/3+120ln2=-60+120×0.7=-60+84=24 kJ So, (2,3,4) are correct.

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1 The magnitude of the total work done in the process $A \to B \to C$ is $144 k J$ .

2 The magnitude of the work done in the process $B \to C$ is $84 k J$ .

3 The magnitude of the work done in the process $A \to B$ is $60 k J$ .

4 The magnitude of the work done in the process $C \to A$ is zero.