📂Thermal Physics

Heat Capacity

Definition¹

The heat $dQ$ required to raise the temperature of an object by $dT$ is called the heat capacity, and it is denoted as follows by following the letter C of capacity.

$$ C = \dfrac{dQ}{dT} [\text{J/K}] $$

Explanation

Particularly in physics, the heat capacity per unit mass, called specific heat capacity, is not very important. Thermodynamics is more interested in the phenomena that occur in a system in general rather than the characteristics of a specific material.

In fact, it is more intuitive to think about the reciprocal of heat capacity, $\dfrac{1}{C} = \dfrac{dT}{dQ}$. A small heat capacity $C$ means that $\dfrac{1}{C}$ is large, which implies a large temperature change with a change in thermal energy.

If you still don’t understand, consider the following analogy.

Imagine filling two basins with a bottom area of $C_{1}$ and $C_{2}$ respectively with the same amount of water, $Q$. Even with the same amount of water, the basin with a larger bottom area will have a lower height, and the basin with a smaller bottom area will have a higher height.

If you actually pour the water, the depth will differ as $T_{1}$ and $T_{2}$ as shown above. Applying them directly to the definition of heat capacity, one can see that it perfectly fits. Just as a basin with a larger bottom area can store more water at the same height, a system with a larger heat capacity can store more thermal energy at the same temperature.

Meanwhile, the heat capacity at constant volume is denoted by $C_{V}$, and the heat capacity at constant pressure is denoted by $C_{p}$. Common sense indicates that $C_{p}$ is larger than $C_{V}$ because if the volume is not constant, the movement of gas molecules is also considered, so the energy change is greater (of course, this is a qualitative explanation, so even if it makes sense, it should not be trusted, and it is not a problem even if it doesn’t make sense).

Values obtained through actual experiments are also close to $\displaystyle C_{p} = {{5} \over {2}} R > {{3} \over {2} } R = C_{V}$, and the same results can be obtained theoretically as well.