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Chapter 7: Problem 920

The specific heat at constant pressure and at constant volume for an ideal gas are \(\mathrm{C}_{\mathrm{p}}\) and \(\mathrm{C}_{\mathrm{v}}\) and adiabetic \(\&\) isothermal elasticities are \(E_{\Phi}\) and \(E_{\theta}\) respectively. What is the ratio of \(\mathrm{E}_{\Phi}\) and \(\mathrm{E}_{\theta}\) (A) \(\left(\mathrm{C}_{\mathrm{v}} / \mathrm{C}_{\mathrm{p}}\right)\) (B) \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)\) (C) \(\mathrm{C}_{\mathrm{p}} \mathrm{C}_{\mathrm{y}}\) (D) \(\left[1 / C_{p} C_{y}\right]\)

Short Answer

Expert verified
The ratio of adiabatic elasticity (E_F) and isothermal elasticity (E_θ) is given by: \(\frac{E_F}{E_θ} = \frac{C_p}{C_v}\) Hence, the correct answer is (B) \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)\).

Step by step solution

01

1. Recall the relationship between Cp, Cv, and the adiabatic index

We know that the adiabatic index, denoted by γ (gamma), is given by the ratio of specific heat capacities: \(γ = \frac{C_p}{C_v}\)
02

2. Recall the expression for adiabatic elasticity (E_F)

The adiabatic elasticity, E_F, is related to the adiabatic index as follows: \(E_F = γ\)
03

3. Recall the expression for isothermal elasticity (E_θ)

Isothermal elasticity, E_θ, is given by: \(E_θ = 1\)
04

4. Find the ratio E_F / E_θ

Now, we can find the ratio of adiabatic elasticity to isothermal elasticity: \(\frac{E_F}{E_θ} = \frac{γ}{1} = \frac{C_p}{C_v}\) So, the correct answer is (B) \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)\).

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