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Chapter 7: Problem 921

What is the ratio of the adiabatic to isothermal elasticities of a triatomic gas ? (A) \((3 / 4)\) (B) \((4 / 3)\) (C) 1 (D) \((5 / 3)\)

Short Answer

Expert verified
The ratio of the adiabatic to isothermal elasticities for a triatomic gas is \( \frac{7}{5} \). However, this option is not available in the given choices. There may be an error in the question or the options.

Step by step solution

01

Find the Degrees of Freedom for Triatomic Gas

A triatomic gas has 3 translational and 2 rotational degrees of freedom. Therefore, it has a total of 3 + 2 = 5 degrees of freedom.
02

Calculate Heat Capacity at Constant Volume (Cv)

We calculate Cv using the formula: Cv = (f/2)R, where f is the degrees of freedom, and R is the gas constant. Here, f = 5. Cv = \(\frac{5}{2}\)R
03

Calculate Heat Capacity at Constant Pressure (Cp)

We calculate Cp using the formula: Cp = Cv + R. Cp = \(\frac{5}{2}\)R + R = \(\frac{7}{2}\)R
04

Calculate Adiabatic Elasticity (Cp / Cv)

The adiabatic elasticity is the ratio of Cp to Cv: Adiabatic Elasticity = \( \frac{Cp}{Cv} = \frac{\frac{7}{2}R}{\frac{5}{2}R} \)
05

Simplify the Adiabatic Elasticity

Cancel out the common factor R and simplify: Adiabatic Elasticity = \( \frac{7}{5} \)
06

Calculate Isothermal Elasticity

Isothermal elasticity is always equal to 1 for an ideal gas.
07

Find the Ratio of Adiabatic to Isothermal Elasticity

The ratio of the adiabatic to isothermal elasticity is \( \frac{7}{5} \) divided by 1: Ratio = \( \frac{7}{5} \) Comparing the given options, the answer is not available in the list. There may be an error in the question or the options.

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