An ideal gas (C_{P}/C_{V} = γ) is taken through a process in which the pressure and the volume vary as p = αV^{b}. Find the value of b for which the specific heat capacity in the process is zero.

**Given:**

p = aV^{b}

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

**Formula used:**

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 = nC_{v}dT, we get

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

heat capacity, and C_{v} = specific heat capacity at constant volume,

dT = change in temperature,

Since specific heat capacity is 0(given),

…(iii)

(after integration of from V_{1} to V_{2})

Now, from equation of state, PV = nRT,

Where

P = pressure,

V = volume,

n = number of moles,

R = universal gas constant,

T = temperature.

Substituting p = aV^{b} from (i):

aV^{b+1} = nRT

=> … (iv)

Substituting (iv) in (iii),

(Since (T_{2} - T_{1}) = dT)

=>

=> b = -

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