Question

The ratio of specific heats 

\((\frac{c_p}{c_v}) \)

in terms of degree of freedom (f) is given by:

• A. \((1+\frac{f}{3})\)
• B. \((1+\frac{2}{f})\)
• C. \((1+\frac{f}{2})\)
• D. \((1+\frac{1}{f})\)

Answer 1

The Correct Option is A

To solve the problem regarding the ratio of specific heats in terms of the degree of freedom, we need to refer to the fundamental concepts of thermodynamics related to gases.

In thermodynamics, the ratio of specific heats, denoted by \(\gamma\), is defined as:

\(\gamma = \frac{c_p}{c_v}\)

where:

• \(c_p\) is the specific heat at constant pressure
• \(c_v\) is the specific heat at constant volume

For a gas with a degree of freedom f, the specific heat capacities are related to the universal gas constant \(R\) by the relations:

• \(c_v = \frac{f}{2}R\)
• \(c_p = \left(\frac{f}{2} + 1\right)R\)

The ratio \(\gamma\) can thus be expressed as:

\[\gamma = \frac{c_p}{c_v} = \frac{\left(\frac{f}{2} + 1\right)R}{\frac{f}{2}R}\]

Cancelling the \(R\) in the numerator and denominator, we get:

\[\gamma = \frac{\frac{f}{2} + 1}{\frac{f}{2}} = \frac{f + 2}{f}\]

Simplifying the expression further, we can divide both terms by \(f\):

\[\gamma = 1 + \frac{2}{f}\]

Conclusion: By comparing this derived expression with the options given, it corresponds to:

\((1+\frac{2}{f})\)

Hence, the correct answer is \((1+\frac{2}{f})\), not the initially provided correct answer \((1+\frac{f}{3})\). The discrepancy suggests a misstatement or misunderstanding of the solution structure in the options list.