The ratio of the molar heat capacities of an ideal gas is C_{P}/C_{V} = 7/6. Calculate the change in internal energy of 1.0 mole of the gas when its temperature is raised by 50 K

(a) keeping the pressure constant,

(b) keeping the volume constant and

(c) adiabatically.

**Given:**

n = number of moles = 1,

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

C_{p} = specific heat capacity at constant pressure

dT = change in temperature = 50K.

γ= Ratio of molar heat capacities = C_{P}/C_{V} = 7/6 => C_{v} = 6C_{p}/7.

(a) **Formula used:**

Pressure constant: Isobaric process. For an isobaric process,

change in internal energy dU = nC_{v}dT,

Where

n = number of moles,

C_{v} = specific heat at constant volume,

dT = rise in temperature

Also, C_{p}-C_{v} = R.

C_{p} = specific heat at constant pressure

C_{v} = specific heat at constant volume

R = universal gas constant = 8.314 J/mol/K

Substituting: C_{p} - 6C_{p}/7 = C_{p}/7 = R => C_{p} = 7R.

Therefore C_{v} = C_{p} - R = 6R = (6 X 8.314)J/mol/K

Therefore,

dU = 1 mol X (6 X 8.314)J/mol/K X 50K = 2494.2 J(Ans)

(b) Volume constant: Isochoric process, dV = 0(change in volume)

First law of thermodynamics gives us: dU = dQ - dW

Where dU = change in internal energy, dQ = change in heat,

dW = work done = Pressure x change in volume = PdV

Since dV = 0, dU = dQ.

Hence, dU = nC_{v}dT since dQ = nC_{v}dT.

Where

n = Number of moles,

C_{v} = Specific heat at constant volume,

dT =Change in temperature

Putting the values in the above formula, we get

Therefore, dU = 1 mol x (6 x 8.314) J/mol/K x 50K

= 2494.2 J(Ans)

(c) Adiabatic process: dQ(heat change) = 0. Therefore,

dU(change in internal energy) = dW(work done)

Since dQ = dU + dW.

For an adiabatic process, dW = dT/(

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