The equation Cp - Cv = R is a relationship derived from the laws of thermodynamics and the ideal gas law. Cp and Cv represent the specific heat capacities at constant pressure and constant volume, respectively, and R is the gas constant. The equation can be proven using the first and second laws of thermodynamics and the ideal gas law. Here's a step-by-step explanation:
Start with the first law of thermodynamics for an ideal gas: ΔU = q - w,
where ΔU is the change in internal energy, q is the heat added to the gas, and w is the work done by the gas.
Express the work done by the gas using the ideal gas law: w = PΔV,
where P is the pressure and ΔV is the change in volume.
Rewrite the first law of thermodynamics using the ideal gas law: ΔU = q - PΔV.
Assume a constant volume process, where ΔV = 0. This means that no work is done by the gas, and the first law simplifies to: ΔU = qv, (where qv represents heat at constant volume).
Now, consider the heat added at constant pressure, which we'll denote as qp. For a constant pressure process, the first law becomes: ΔU = qp - PΔV.
Use the ideal gas law to rewrite PΔV in terms of the initial and final states of the gas: PΔV = nRΔT,
where n is the number of moles of gas, R is the gas constant, and ΔT is the change in temperature.
Substitute the expression for PΔV into the equation from step 5: ΔU = qp - nRΔT.
Rearrange the equation by dividing both sides by nΔT: (ΔU / nΔT) = (qp / nΔT) - R.
Recognize that ΔU / nΔT is the definition of the specific heat capacity at constant volume, Cv: Cv = (qp / nΔT).
Finally, substitute Cv into the equation from step 8: Cv = Cp - R.
Thus, the equation Cp - Cv = R is proven based on the assumptions of the ideal gas law, the first law of thermodynamics, and the concept of specific heat capacities at constant volume and constant pressure.