Given:

P = kV …

Where,

P = pressure,

V = volume,

k = constant.

Formula used:

Equation of state of ideal gas:

PV = nRT = constant … (ii)

Where,

n is the number of moles of the gas,

R is the gas constant,

T is the temperature,

P = pressure, V²

T = temperature.

From (i), multiplying by dV on both sides:

PdV = kVdV.

Integrating from V = V1 to V2, we get

= with lower limit V₁ and upper limit V₂

=

Now, we know, PV = nRT - equation of state,

Where P = pressure, V = volume, n = number of moles, R = universal gas constant, T = temperature

Hence we can write, V₁ = nRT₁/P₁. Since P₁ = kV₁, this becomes:

kV₁² = nRT₁. Similarly, KV₂² = nRT₂,

P₁, V₁, T₁ - Pressure, volume, temperature of first gas

P₂, V₂, T₂ - Pressure, volume, temperature of second gas

Therefore, subsituting, the above equation becomes:

= = … (iii)

Now, => (since P = kV) … (iv)

But Q = U + ഽPdV (first law of thermodynamics), where Q = heat, U = change in internal energy, W = total work done = ഽPdV

=> nCdT = nC_vdT + (nR/2)dT

(since Q = nCdT and U = nC_vdT)

=> C = C_v + nR/2 (proved),

Where

n = number of moles,

C = specific heat capacity,

C_v = specific heat capacity at constant volume,

R = universal gas constant,

dT = rise in temperature.