An ideal gas undergoes a quasi-static, reversible process in which its molar heat capacity C remains constant. If during this process the relation of pressure p and volume V is given by `pV^(n)` = constant, then n is given by (Here `C_(p) and C_(v)` are molar specific heat at constant pressure and constant volume, respectively)

Options

A. `n = (C_p)/(C_v)`
B. `n = (C - C_p)/(C- C_v)`
C. `n = (C_p- C)/(C- C_v)`
D. `n = (C - C_v)/(C - C_p)`

Answer

B

Explanation

Here, `pV^(n) = k` [constant] ……(1)
For 1 mol of an ideal gas,
`pV = RT` …..(2)
By `(1) -: (2)` we get, `V^(n-1) T = k/R`
`:. ((dV)/(dT)) = -(V)/((n-1)T) = V/((1-n)T)`
According to the first law of thermodynamics,
`dQ = C_(v)dT + pdV`
`:. (dQ)/(dT) = C_(v) + p((dV)/(dT)) = C_(v) + (pV)/((1-n)T) = C_(v) + R/(1-n)`
So, heat capacity, `C = C_(v)+R/(1-n)`
or, `1 - n= (R)/(C-C_v)`
or, `n = 1 - R/(C - C_v) = (C-(C_v + R))/(C - C_v) = (C - C_p)/(C - C_v)`
`[ :. C_(p) - C_(v) = R]`.