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Question
Prove the Mayer's relation `C_p - C _v = R/J`
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Solution
Consider one mole of an ideal gas that is enclosed in a cylinder by a light, frictionless, airtight piston.
Let P, V, and T be the pressure volume and temperature of the gas, respectively. If the gas is heated so that its temperature rises by dT, but volume remains constant.
Let dQ1 → be the amount of heat supplied to the gas,
dE → be the increase in internal energy. Since, volume of gas is constant, no work is done in moving the piston.
dQ1 = dE + dW
∴ dQ1 = dE = CVdT ...(i)
(As dW = 0)
If the gas is heated to the same temperature at constant pressure.
Let dV → be the increase in volume.
Therefore work is done and it is given by
dW = PdV
Let dQ2 → be the heat supplied to the gas
∴ dQ2 = dE + dW
∴ CpdT = dE + dW ...(ii)
Putting dE = CvdT and dW = PdV
∴ Equation (ii) becomes
CPdT = CvdT + PdV
(Cp − Cv)dT = PdV ...(iii)
For one mole of gas
PV = RT
∴ pdV = RdT
∴ (Cp − Cv)dT = RdT
Cp − Cv = R ...(iv)
This is known as Mayer's relation.
Generally, heat supplied is measured in calories, and work done is measured in joules. Therefore, equation (iv) is modified as follows:.
`C_p - C _v = R/J`
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