What is an intuitive explanation for the change in internal energy being equal to the heat capacity times the change in temperature? I read that the change in internal energy, $\Delta U$, is equal to the heat capacity at constant volume, $C_V$, times the change in temperature, $\Delta T$. So, in other words, we have that $\Delta U = C_V \Delta T$. Wikipedia defines internal energy as the energy contained within a thermodynamic system, and defines heat capacity as the amount of heat to be supplied to a given mass of a material to produce a unit change in its temperature. As a novice, I want to understand $\Delta U = C_V \Delta T$ intuitively. What is an intuitive explanation for $\Delta U = C_V \Delta T$? The point of intuition that I'm having trouble with is the idea that the change in internal energy is equal to the heat capacity times the change in temperature; specifically, in an intuitive sense, this seems like overkill ($C_V \Delta T$ seems like it would be too much to equal the change in internal energy).
 A: By definition, heat capacity at constant volume is the amount of heat supplied to a system per unit change in its temperature, assuming its volume remains the same. Mathematically, if $dQ_V$ is an infinitesimal amount of heat the system absorbs under constant volume, which causes a corresponding change in temperature $dT$ of the system, then
$$C_V = \frac{dQ_V}{dT}.$$ In general (constant volume or not), we assume that the heat supplied, $dQ$, can cause only two things:
a) It can change the internal energy, $U$, of the system; and,
b) It can do work, $W$, on the system.
Now, here comes a simplification: we only consider "volume-based" work -- i.e., only the kind of work that results in change in volume of the system. For example, if the system is a gas trapped inside a piston, the only relevant kind of work is the piston getting pushed, compressing the gas inside. Of course, other kinds of work are also possible, such as electrical work, chemical work, etc., but we can ignore them for the time being.
Now, from the first law of thermodynamics,
$$dQ = dU + dW,$$ that is, the total heat supplied has to change the internal energy of the system, or it has to do work on the system, or both. This is just the law of energy conservation put into the context of thermodynamics. If the volume of the system does not change (called an isochoric process) by whatever reason, then $dW = 0$ by our restriction of only volume-based work. This means $dQ = dQ_V$: the heat supplied is under constant volume. This leads to
$$dQ_V = dU \implies dU = C_V \, dT$$ using the first equation.
