The derivation of the relation Cp-Cv=R involves considering the change in enthalpy and internal energy of a gas during a constant pressure process and a constant volume process.
For a constant pressure process, the change in enthalpy (ΔH) of a gas is given by:
ΔH = qp = CpΔT
where qp is the heat added to the gas at constant pressure, Cp is the specific heat at constant pressure, and ΔT is the change in temperature of the gas.
For a constant volume process, the change in internal energy (ΔU) of a gas is given by:
ΔU = qv = CvΔT
where qv is the heat added to the gas at constant volume, Cv is the specific heat at constant volume, and ΔT is the change in temperature of the gas.
The total heat added to the gas in both processes is the same, so we can write:
qp = qv
Substituting the expressions for qp and qv in terms of Cp and Cv, and rearranging, we
obtain:
CpΔT - CvΔT = 0
Therefore, we can conclude that:
Cp - Cv = 0
However, this derivation assumes that the gas is ideal and follows the ideal gas law. If we consider a non-ideal gas, there may be some work done by or on the gas during the constant pressure and constant volume processes, which would affect the internal energy and enthalpy changes. In that case, Cp-Cv would not necessarily be zero.
The ratio of specific heats, γ, is defined as the ratio of the specific heat at constant pressure, Cp, to the specific heat at constant volume, Cv:
γ = Cp/Cv
Using the ideal gas law, we can express Cp and Cv in terms of the gas constant, R:
Cp = γR/(γ-1)
Cv = R/(γ-1)
Substituting Cv in the expression for Cp, we obtain:
Cp - Cv = γR/(γ-1) - R/(γ-1)
Simplifying, we obtain:
Cp - Cv = R
Therefore, we can conclude that:
γ = Cp/Cv = (Cp - Cv)/Cv + 1 = R/Cv + 1 = (γ - 1)/γ + 1
This expression relates the ratio of specific heats to the adiabatic index, which is a parameter that describes the compressibility of a gas.
