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Chapter 12: Problem 26

The molar specific heats of an ideal gas at constant pressure and volume are denoted by \(C_{p}\) and \(C_{v}\), respectively. Further, \(\frac{C_{p}}{C_{v}}=\gamma\) and \(R\) is the gas constant for 1 gm mole of a gas. Then \(C_{v}\) is equal to (A) \(R\) (B) \(\gamma R\) (C) \(\frac{R}{\gamma-1}\) (D) \(\frac{\gamma R}{\gamma-1}\)

Short Answer

Expert verified
The short answer is \(C_v = \frac{R}{\gamma - 1}\), which corresponds to option (C).

Step by step solution

01

Write the equation for the gas constant R

We start by noting that the difference in molar specific heat at constant pressure and constant volume is equal to the gas constant. This can be written as: \(C_p - C_v = R\)
02

Use the given relationship between \(C_p\) and \(C_v\)

We are given that the ratio of \(C_p\) to \(C_v\) is equal to \(\gamma\). This can be written as: \(C_p = \gamma C_v\)
03

Substitute the relationship found in Step 2 into the equation in Step 1

Substituting the expression for \(C_p\) in terms of \(C_v\) found in the previous step into the equation in Step 1, we get: \(\gamma C_v - C_v = R\)
04

Solve for \(C_v\) in terms of R and \(\gamma\)

We can arrange the equation found in Step 3 to isolate \(C_v\): \(C_v (\gamma - 1) = R\) Now, we can divide both sides of the equation by \((\gamma - 1)\) to find the value of \(C_v\): \(C_v = \frac{R}{\gamma - 1}\) Therefore, the correct answer is \(C_v = \frac{R}{\gamma - 1}\), which corresponds to option (C).

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Most popular questions from this chapter

One mole of an ideal gas at an initial temperature of \(T K\) does \(6 R\) joules of work adiabatically. If the ratio of specific heats of this gas at constant pressure and at constant volume is \(5 / 3\), the final temperature of gas will be (A) \((T+2.4) \mathrm{K}\) (B) \((T-2.4) \mathrm{K}\) (C) \((T+4) \mathrm{K}\) (D) \((T-4) \mathrm{K}\)

When a system is taken from state \(i\) to state \(f\) along the path iaf, it is found \(Q=50 \mathrm{cal}\) and \(W=20\) cal. Along the path \(i b f, Q=36\) cal. \(W\) along the path \(i b f\) is [2007] (A) \(6 \mathrm{cal}\) (B) \(16 \mathrm{cal}\) (C) \(66 \mathrm{cal}\) (D) \(14 \mathrm{cal}\)

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Which one of the following statements is incorrect? (A) If positive work is done by a system in a thermodynamic process, its volume must increase. (B) If heat is added to a system, its temperature must increase. (C) A body at \(20^{\circ} \mathrm{C}\) radiates in a room, where room temperature is \(30^{\circ} \mathrm{C}\). (D) If pressure vs temperature graph of an ideal gas is a straight line, then work done by the gas is zero.

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