An ideal gas (CP/CV = γ) is taken through a process in which the pressure and the volume vary as p = αVb. Find the value of b for which the specific heat capacity in the process is zero.


p = aVb

p = pressure, V = volume, a and b are constants.

Formula used:

We know, Q = U + pdV from first law of thermodynamics,

where Q = change in heat, U = change in internal energy and pdV = W = total work done, p = pressure, V = volume.

Since Q = nCdT, and U = nCvdT, we get

… (ii), n = no. of moles, C = specific

heat capacity, and Cv = specific heat capacity at constant volume,

dT = change in temperature,

Since specific heat capacity is 0(given),


(after integration of from V1 to V2)

Now, from equation of state, PV = nRT,


P = pressure,

V = volume,

n = number of moles,

R = universal gas constant,

T = temperature.

Substituting p = aVb from (i):

aVb+1 = nRT

=> … (iv)

Substituting (iv) in (iii),

(Since (T2 - T1) = dT)


=> b = -