Derive the Mayer's relation between for molar as well as principal specific heats?
Answers
Answer:
Mayer's formula is Cp - Cv = R. Here Cp is molar specific heat capacity of an ideal gas at constant pressure, Cv is its molar specific heat at constant volume and R is the gas constant.
Consider a cylinder with a volume of V and a piston with an area of A that is filled with n moles of an ideal gas at a pressure of P.
Consider heating the gas at a fixed pressure, which increases its temperature by dT. The piston moves outward at a very tiny distance (dx) due to the overall force F = PA that the gas applies to it.
dW = Fdx = PAdx = PdV … (1)
where Adx = dV represents the increase in gas volume caused by the expansion. The work the gas does on its surroundings as a result of expanding is measured in units of dW. According to the first rule of thermodynamics, dQp = dE + dW=dE + PdV if the gas is heated by dQp and its internal energy increases by dE.
dQ = nCp dT, if Cp is the gas's molar specific heat capacity at constant pressure.
∴ nCpdT = dE + PdV. (2)
On the contrary, if the gas were heated from its initial state at constant volume (rather than constant pressure) so that its temperature increased by the same amount dT, then dW=0.
Since an ideal gas' internal energy solely depends on temperature, the increase in internal energy would once more be dE. According to the first law of thermodynamics and the definition of molar-specific heat capacity at constant volume, dQv would be the heat delivered to the gas in this scenario (C_v)
dQv=dE=nCvdT …..(3)
from (2) and (1)
nCpdT= nCvdT+PdV
therefore,
Cp-Cv = PdV/ndT
For an ideal gas,
PV=Nrt
At constant pressure,
PdV=nRdT
Therefore,
dV/dT= nR/P ……..(5)
From (4) & (5)
Cp – Cv = P/n, nR/P =R …..(6)
This is Mayer's relation between Cp and Cv
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