\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)
Chapter 7: Thermodynamics
3.

Heat Capacity

heat capacity
\(C\)
J/K
It’s probably not surprising to you that when heat flows into an object, its temperature increases. However, the amount by which the temperature increases depends on the object. For example, a stove can increase the temperature of a metal pot fairly quickly, but if that pot is filled with water than it takes a lot more heat, and so a lot longer. The resistance of an object to having its temperature raised is called its heat capacity $C$, which has units of J/K. The relevant formula is
$$\Delta T={Q\over C}$$

The temperature and the heat capacity are inversely proportional, so for the same amount of heat, a higher heat capacity results in a smaller change in temperature. (One might say that an object with more heat capacity has “more room” to store the heat without having to react by getting warmer.)

Heat capacity depends in part on the mass of an object: it takes much less heat to boil a cup of water than to boil a lake. In fact, the heat capacity is proportional to the mass, and we can write it as

$$C=mc$$

specific heat
\(c\)
J/kg/K
where the specific heat $c$ (lower-case, measured in J/kg/K) depends on what kind of material the object is made of.

For example, iron has a specific heat of 449 J/kg/K, because it takes $449\u{J}$ of heat to increase 1 kg of iron by 1K (or 1C°) of temperature. Liquid water's specific heat is more than nine times larger, and so it takes a lot more heat to warm water, and water takes longer to cool off as well. This is why coastal regions tend to have more moderate climates than interior ones: the nearby ocean is able to absorb a lot of heat from the air in the summertime, keeping it cooler, and during the winter, the ocean releases that heat, warming the air.

A thermal reservoir is an object whose heat capacity is so large that its temperature barely changes at all as it absorbs or released heat. The atmosphere or the ocean might be considered reservoirs for some applications: as coffee cools it dumps heat into the air, but the air’s temperature won’t change appreciably because of it.

Thermal Energy
\(E_{th}\)
J
We can also use the heat capacity to specify the total amount of thermal energy in an object, which is

$$E_{th}=CT=mcT$$

where $T$ is measured in Kelvin.