A sample of an ideal gas (γ = 1.5) is compressed adiabatically from a volume of 150 cm^{3} to 50 cm^{3}. The initial pressure and the initial temperature are 150 kPa and 300 K. Find

(a) the number of moles of the gas in the sample,

(b) the molar heat capacity at constant volume,

(c) the final pressure and temperature,

(d) the work done by the gas in the process and

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

Given P1 = 150 KPa = 150 × 10^{3} Pa, V1 = 150 cm^{3} ,V2=50 cm^{3}, T1 = 300 K

(a)By ideal gas equation,

where P, V and T are the pressure, volume and absolute temperature; n is the number of moles of gas; R is the ideal gas constant

I.e.

(b)We know Cp/Cv=γ R=Cp-Cv,

where the molar heat capacity C, at constant pressure, is represented by Cp, at constant volume, the molar heat capacity C is represented by Cv

Now,

(c) Since the process is adiabatic,

PV^{γ} = const.

I.e.

Also, as the process is adiabatic,

(d) From the first law of thermodynamics,

, where dQ is the amount of heat supplied which is zero in an adiabatic process.

i.e.

, where n is the number of moles, Cv is the molar heat capacity at constant volume and dT is the change in temperature

(e)Change in internal energy, dU is

, where n is the number of moles, Cv is the molar heat capacity at constant volume and dT is the change in temperature.

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