The amount of heat needed to raise the temperature of 1 mole of a substance by one Celsius degree (or, equivalently, one kelvin) is called the molar heat capacity of the system, denoted by the letter C. If a small amount of heat dQ is put into n moles of a substance, and the resulting change in temperature for the system is dT, then

C=1ndQdT.

This is the definition of molar heat capacity--the amount of heat Q added per infinitesimal change in T, per mole.

A heated gas tends to expand, and the heat capacity depends on whether the gas is held at constant volume or allowed to expand so that it remains at constant pressure. The molar heat capacities are denoted Cv and Cp for constant volume and constant pressure, respectively.

When a gas is held at constant volume and heated, dV=0, so the work, pdV done by the gas is zero. The first law of thermodynamics, dQ=dU+dW, therefore implies that all of the heat must go into increasing the gas's internal energy U.

When a gas is held at constant pressure as it is heated, the same amount of heat is required to increase the gas's internal energy. In addition, the gas expands and does work. The first law therefore demands addition of extra heat to do the work. Consequently, for a gas held at constant pressure, it takes more heat for a given increase in temperature than it does for a gas held at constant volume.

This problem concerns the important relationship between molar heat capacity at constant volumeCv and the molar heat capacity at constant pressure Cp.

This discussion shows that the molar heat capacity of a gas depends on the circumstances under which the gas is heated. The molar heat capacity for a gas held at constant pressure is greater than that for a gas held at constant volume. In this problem, you will derive a single equation that relates Cp and Cv for an ideal gas.

A) Consider an ideal gas being heated at constant volume, and let Cv be the gas's molar heat capacity at constant volume. If the gas's infinitesimal change in temperature is dT, find the infinitesimal change in internal energy dU of n moles of gas.

Express the infinitesimal change in internal energy in terms of given quantities.

B) Now suppose the ideal gas is being heated while held at constant pressure p. The infinitesimal change in the gas's volume is dV, while its change in temperature is dT. Find the gas's molar heat capacity at constant pressure, Cp.

Express Cp in terms of some or all of the quantities Cv, p, dV, n, and dT.

C) Suppose there are n moles of the ideal gas. Simplify your equation for Cp using the ideal gas equation of state: pV=nRT.

Express Cp in terms of some or all of the quantities Cv, n, and the gas constant R.

D) The ratio of the specific heats Cp/Cv is usually denoted by the Greek letter γ. For an ideal gas, find γ.

Give your answer in terms of some or all of the quantities n, R, and Cv.