Final answer:
The question is about deriving the heat capacity equation for an ideal gas at constant volume. Using the first law of thermodynamics and the ideal gas law, we can relate the change in internal energy to the heat capacity at constant volume (Cv) and constant pressure (Cp).
Step-by-step explanation:
The student is asking for the derivation of the equation for the heat capacity of an ideal gas, where specific heat capacity is measured under constant volume conditions. According to the hint provided, it seems the solution should also involve recognizing that internal energy (U) is a state function, and using the property of the exact differential dU.
In thermodynamics, the heat capacity at constant volume (Cv) for an ideal monatomic gas is expressed as Cv = (d/2)R, where d represents the number of degrees of freedom of the gas molecules, and R is the universal gas constant. When calculating the molar heat capacity at constant pressure (Cp), we use the relationship Cp = Cv + R. This equation is derived from the ideal gas law and is a good approximation for dilute gases including monatomic, diatomic, and polyatomic.
In a scenario where the process is isochoric, which means the volume is held constant, no work is done and the first law of thermodynamics simplifies to dEint = dQ, where dEint is the infinitesimal change in internal energy and dQ is the infinitesimal heat added to the system. This can also be written as dQ = CvndT, highlighting that at constant volume, the amount of heat added to the system is directly proportional to the temperature change and the molar heat capacity under constant volume.