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.