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Concepts Of Physics MCQ Edition [Volume 2]
Exam
Concepts Of Physics MCQ Edition [Volume 2]
Subject
Physics
Chapter
Laws of Thermodynamics

Concepts Of Physics MCQ Edition [Volume 2] · Physics · Laws of Thermodynamics

As shown in the figure, a gas can be taken from state A to state B...

As shown in the figure, a gas can be taken from state AA to state BB via three different paths. Determine the work done by the gas in each of the three paths (ABAB, ACBACB, and ADBADB respectively).
As shown in the figure, a gas can be taken from state A to state B via three different paths. Determine the work done

Options

  1. A0.30 J0.30\text{ J}, 0.15 J0.15\text{ J}, and 0.45 J0.45\text{ J}
  2. B0.35 J0.35\text{ J}, 0.20 J0.20\text{ J}, and 0.50 J0.50\text{ J}
  3. C0.30 J0.30\text{ J}, 0.45 J0.45\text{ J}, and 0.15 J0.15\text{ J}
  4. D0.45 J0.45\text{ J}, 0.30 J0.30\text{ J}, and 0.15 J0.15\text{ J}

Correct answer

C. 0.30 J0.30\text{ J}, 0.45 J0.45\text{ J}, and 0.15 J0.15\text{ J}

Step-by-step solution

From the given V−pV-p graph, the coordinates of the states are:
pA=10 kPap_A = 10 \text{ kPa}, VA=10 ccV_A = 10 \text{ cc}
pB=30 kPap_B = 30 \text{ kPa}, VB=25 ccV_B = 25 \text{ cc}
pC=30 kPap_C = 30 \text{ kPa}, VC=10 ccV_C = 10 \text{ cc}
pD=10 kPap_D = 10 \text{ kPa}, VD=25 ccV_D = 25 \text{ cc}

The work done by the gas is given by W=∫p dVW = \int p \, dV, which is the area under the curve projected onto the volume (VV) axis.

For path ABAB:
The work done is the area of the trapezium formed by the line ABAB and the VV-axis.
WAB=12(pA+pB)(VB−VA)W_{AB} = \dfrac{1}{2} (p_A + p_B) (V_B - V_A)
WAB=12(10+30)×103×(25−10)×10−6W_{AB} = \dfrac{1}{2} (10 + 30) \times 10^3 \times (25 - 10) \times 10^{-6}
WAB=12×40×103×15×10−6=0.30 JW_{AB} = \dfrac{1}{2} \times 40 \times 10^3 \times 15 \times 10^{-6} = 0.30 \text{ J}

For path ACBACB:
The process ACAC is isochoric (V=constantV = \text{constant}), so WAC=0W_{AC} = 0.
The process CBCB is isobaric (p=constantp = \text{constant}), so WCB=pC(VB−VC)W_{CB} = p_C (V_B - V_C).
WCB=30×103×(25−10)×10−6=30×15×10−3=0.45 JW_{CB} = 30 \times 10^3 \times (25 - 10) \times 10^{-6} = 30 \times 15 \times 10^{-3} = 0.45 \text{ J}
WACB=WAC+WCB=0+0.45=0.45 JW_{ACB} = W_{AC} + W_{CB} = 0 + 0.45 = 0.45 \text{ J}

For path ADBADB:
The process ADAD is isobaric (p=constantp = \text{constant}), so WAD=pA(VD−VA)W_{AD} = p_A (V_D - V_A).
WAD=10×103×(25−10)×10−6=10×15×10−3=0.15 JW_{AD} = 10 \times 10^3 \times (25 - 10) \times 10^{-6} = 10 \times 15 \times 10^{-3} = 0.15 \text{ J}
The process DBDB is isochoric (V=constantV = \text{constant}), so WDB=0W_{DB} = 0.
WADB=WAD+WDB=0.15+0=0.15 JW_{ADB} = W_{AD} + W_{DB} = 0.15 + 0 = 0.15 \text{ J}

The work done in paths ABAB, ACBACB, and ADBADB are 0.30 J0.30 \text{ J}, 0.45 J0.45 \text{ J}, and 0.15 J0.15 \text{ J} respectively.

Answer: 0.30 J0.30\text{ J}, 0.45 J0.45\text{ J}, and 0.15 J0.15\text{ J}
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