Final answer:
The formula dU = nCvdT is derived from the first law of thermodynamics and the fact that internal energy of an ideal gas only depends on temperature. It is valid for all thermodynamic processes involving a change in temperature, and it simplifies to this form at constant volume since no work is involved.
Step-by-step explanation:
The formula dU = nCvdT is derived based on the thermodynamic principle that internal energy change dU for a system at constant volume is equal to the heat added to the system dQ, since no work is done (W = 0).
As internal energy is a state function that only depends on temperature for an ideal gas, the change in internal energy can be expressed as the product of the amount of substance n in moles, the molar specific heat at constant volume Cv, and the infinitesimal change in temperature dT.
Using the first law of thermodynamics, we see that dU = dQ - dW. At constant volume, work done is zero (W = 0), simplifying the relationship to dU = dQ. This is the case for any thermodynamic process involving an ideal gas where temperature changes, regardless of the process path or the constant-volume condition.
Furthermore, for an ideal gas, the molar heat capacity at constant pressure Cp is related to Cv by the equation Cp = Cv + R, where R is the universal gas constant. This relationship further solidifies the connection between heat capacities and internal energy changes for ideal gases.