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Chapter 12: Problem 21

Three samples of the same gas \(A, B\), and \(C(\gamma=3 / 2)\) have initial equal volumes. Now the volume of each sample is doubled. The process is adiabatic for \(A\), isobaric for \(B\), and isothermal for \(C\). If the final pressures are equal for all three samples, the ratio of their initial pressures is (A) \(2 \sqrt{2}: 2: 1\) (B) \(2 \sqrt{2}: 1: 2\) (C) \(\sqrt{2}: 1: 2\) (D) \(2: 1: \sqrt{2}\)

Short Answer

Expert verified
The ratio of the initial pressures for gas samples A, B, and C is \(2 \sqrt{2}:1:2\), which corresponds to option B.

Step by step solution

01

Write down the Ideal Gas Law

The ideal gas law is given by: \(PV = nRT\), where P is pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
02

Adiabatic Process for gas A

For an adiabatic process, we have: \(PV^{\gamma} = constant\), where \(\gamma\) is the heat capacity ratio. For gas A, \(\gamma = 3/2\). Since the volume doubled, we have \(P_iv^{3/2} = P_fv_f^{3/2}\) which simplifies to: \(P_i = 2^{3/2}P_f\), where \(P_i\) and \(P_f\) represent the initial and final pressures, respectively.
03

Isobaric Process for gas B

For an isobaric process, the pressure remains constant. Since the volume doubled, the final pressure is the same as the initial pressure, \(P_f = P_i\).
04

Isothermal Process for gas C

For an isothermal process, the temperature remains constant. From the ideal gas law, we have: \(PV = nRT\), since \(nRT = constant\), we get \(P_iv_i = P_fv_f\). For doubled volume, this equation simplifies to: \(P_i = 2P_f\).
05

Determine the ratio of initial pressures

We can express the initial pressures for A, B, and C as the final pressure values using the results from Steps 2, 3, and 4: Gas A: \(P_{Ai} = 2^{3/2}P_f\) Gas B: \(P_{Bi} = P_f\) Gas C: \(P_{Ci} = 2P_f\) Taking the ratio of the initial pressures A : B : C, we have: \(P_{Ai}:P_{Bi}:P_{Ci} = 2^{3/2}P_f:P_f:2P_f\) Dividing by \(P_f\), we get: \(P_{Ai}:P_{Bi}:P_{Ci} = 2^{3/2}:1:2\) Thus, the ratio of the initial pressures is 2獗�2:1:2, which corresponds to option B.

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Most popular questions from this chapter

A Carnot engine, whose efficiency is \(40 \%\), takes heat from a source maintained at a temperature of \(500 \mathrm{~K}\). It is desired to have an engine of efficiency \(60 \%\). Then, the intake temperature for the same exhaust (sink) temperature must be: [2012] (A) Efficiency of Carnot engine cannot be made larger than \(50 \%\) (B) \(1200 \mathrm{~K}\) (C) \(750 \mathrm{~K}\) (D) \(600 \mathrm{~K}\)

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Consider two containers \(A\) and \(B\) containing identical gases at the same pressure, volume, and temperature. The gas in container \(A\) is compressed to half of its original volume isothermally while the gas in container \(B\) is compressed to half of its original value adiabatically. The ratio of final pressure of gas in \(B\) to that of gas in \(A\) is (A) \(2^{\gamma-1}\) (B) \(\left(\frac{1}{2}\right)^{\gamma-1}\) (C) \(\left(\frac{1}{1-\gamma}\right)^{2}\) (D) \(\left(\frac{1}{\gamma-1}\right)^{2}\)

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