An ideal gas expands from 100 cm^{3} to 200 cm^{3} at a constant pressure of 2.0 × 10^{5} Pa when 50J of heat is supplied to it. Calculate

(a) the change in internal energy of the gas.

(b) the number of moles in the gas if the initial temperature is 300K.

(c) the molar heat capacity C_{P} at constant pressure and

(d) the molar heat capacity C_{V} at constant volume.

**Given:**

(a) Pressure(P) = 2.0 × 10^{5} Pa, dV(change in volume) = (200-100) cm^{3} = 10^{-4}m^{3}, since 1 m^{3} = 10^{6} cm^{3}

Heat(dQ) = 50 J.

**Formula used:**

Now we know, dQ = dU(change in internal energy) + dW(work)=

dU + PdV (first law of thermodynamics),

Where P = pressure, dV = change in volume.

=> dU = dQ - PdV = (50 - (2.0 × 10^{5} x 10^{-4})) J = 30 J (Ans)

(b) For constant pressure, from equation of state PV/T = constant,

Where P = pressure, V = volume, T = temperature.

Hence we get: =, where

V_{1}(initial volume) = 100 cm^{3}, V_{2}(final volume) = 200 cm^{3}, T_{1} = 300K

=> .

Therefore, PdV = nRdT (for more than one mole),

Where P = pressure, dV = change in volume, n = number of moles, R = universal gas constant = 8.314 J/kg/mol, dT = change in temperature.

=> 2.0 × 10^{5} x 10^{-4} = n x 8.314 x 300 (since T2-T1 = dT = 300K)

Therefore, n = 20/(R x 300) = 0.008 mol (Ans).

(c) dQ(heat) = 50 = nC_{p}dT (at constant pressure),

Where n = number of moles, C_{p} = specific heat at constant pressure, dT = rise in temperature.

=> 50 = 0.008 x C_{p} x 300

=> C_{p} = 20.83 J/mol/K. (Ans)

(d) At constant volume, dU(change in internal energy) = dQ(heat) = nC_{v}dT (since work done dW = PdV = 0, where P = pressure, dV = change in volume), from first law of thermodynamics.

n= number of moles, C_{v} = specific heat capacity at constant volume, dT = change in temperature.

=> 30 = 0.008 x C_{v} x 300

=> C_{v} = 12.5 J/mol/K. (Ans)

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