Question

A certain ideal gas undergoes a polytropic process `PV^(n)` = constant such that the molar specific heat during the process is negative. If the ratio of the specific heats of the gas be `gamma`, then the range of values of `n` will be
A. `0 lt n lt gamma`
B. `1 lt n lt gamma`
C. `n = gamma`
D. `n gt gamma`

Answer 1

Correct Answer - B
b. Since `PV^(n) = ` constant and also `PV = RT`, taking 1 mol of the gas from simplicity
`dU = C_(v) dT`
Where `C_(v) rarr` molar specific heat at constant volume.
Now the molar specific heat in a polytropic process `PV^(n) =` constant is given by
`C_(v) = ((R)/(gamma -1)) - ((R)/(n -1)) = ((n - gamma)R)/((n - 1)(gamma - 1))`
For this equation we see that `C_(v)` will be negative when `n lt gamma` and `n gt 1`, simultaneously i.e., `1 lt n lt gamma`. Since `gamma` for all ideal gases is greater than 1, if `n gt gamma` or `n lt 1`, then `C_(v)` will be positive.