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.
We know, Q = U + ഽpdV from first law of thermodynamics,
where Q = change in heat, U = change in internal energy and ഽpdV = 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 = -