For a gas if ratio of specific heats at constant pressure and volume is \(\gamma\), then value of degrees of freedom is (A) \([2 /(\gamma-1)]\) (B) \([(25) / 2](\gamma-1)\) (C) \([(3 \gamma-1) /(2 \gamma-1)]\) (D) \((9 / 2)(\gamma-1)\)

The degrees of freedom can be defined as "f" and the gas constant as "R". For a gas, the specific heat at constant volume (C_v) can be expressed as: \(C_v = \frac{f}{2}R\) From Mayer's Equation and knowing \(\gamma = \frac{C_p}{C_v}\), we have: \(C_p - C_v = R\) \(C_p = C_v + R\) Now we can write an expression for the specific heat at constant pressure (C_p) in terms of "f" and "R": \(C_p = \frac{f}{2}R + R\)

Since we have the expressions for Cp and Cv in terms of f and R, we can now find an expression for the degrees of freedom "f" in terms of the ratio of specific heats \(\gamma\). \[\gamma = \frac{C_p}{C_v} = \frac{\frac{f}{2}R + R}{\frac{f}{2}R}\] To find the degrees of freedom, we need to isolate "f" in the above equation: First, we can multiply both sides by the denominator \(\frac{f}{2}R\), to get rid of the fraction: \[\gamma \frac{f}{2}R = \frac{f}{2}R + R\] Now, expand the equation to move R terms to one side: \[\gamma f R - 2f R = 2R\] Factor out the fR term to get: \[(\gamma - 2)fR = 2R\] Now, we can solve for f by dividing both sides by the gas constant R and (\(\gamma - 2\)): \[f = \frac{2R}{(\gamma - 2)R}\] Simplify the expression to get the final expression for the degrees of freedom in terms of the given ratio: \[f = \frac{2}{(\gamma - 1)}\] Comparing this expression with the given options, we can conclude that:

The correct option is (A) \([2 /(\gamma-1)]\) as it matches the expression for the degrees of freedom that we derived based on the given ratio of specific heats.