We want to prove that `C_v = ((dU)/(dT))_v` , an expression for heat capacity at constant volume.

Heat capacity is defined as the ratio of the amount of heat added into the system or released by the system resulting to an increase or decrease in temperature. In other words:

`C = Q/(\Delta T)` where Q is heat and T is temperature.

The first law of thermodynamics states that the internal energy is the sum of heat and work:

`dU = Q + W` .

From this, we can get an expression for Q:

`Q = dU - W` .

At constant volume, `\Delta V = 0` . That is, there is no expansion or compression. In thermodynamics, this means that `W = 0` .

Therefore, at constant volume, the heat capacity is:

`C_v = ((\Delta U)/(\Delta T))` .

However, we are concerned with really small changes, at a given condition (constant volume):

`C_v = ((\delta U)/(\delta T))_v` .

Note that subscripts in this case denote the condition that volume is constant.