Physical Chemistry—hotness and coldness
Students often have trouble putting observation into context. So when I teach undergraduate physical chemistry, especially to life science students, it is important to have them understand a process from a physical or conceptual point of view before plunging into a logical (mathematical) description. They have enough problems casting ideas into mathematical equations. The first goal, when introducing a new idea, is to give them a visualization of what is going on. For example, I recently wrote a blog on dispelling the myth that energy is stored in chemical bonds—it is not.
When I start into heat capacity I contrast the temperature of a substance with the feeling of hot and cold. A thermometer will tell you the temperature of a substance, but that does not tell you how much heat is present. If you touch something, you can tell if it is hotter or colder than your hand, but what about two substances at the same temperature?
Suppose outside it is -10 C (14 F) and there you find a piece of steel and a piece of Styrofoam. Which is colder? If you touch the steel it feels colder than the Styrofoam, but they are both at the same temperature. If you placed the steel on the Styrofoam, no heat will flow between them (Third Law of thermodynamics). Since your hand is much hotter than the objects, heat must flow from your hand into them.
The sensation of hotness and coldness does not have to do with the temperature only, there is something else, which is the ability (capacity) of a substance to hold heat. Some hold more (high heat capacity) and some hold less (low heat capacity)
There are three substances here: the steel at -10, the Styrofoam at -10 and your hands, at 37 C. Since heat flows from a hotter to a colder (Second Law), heat flows out of your hands and into the steel or Styrofoam. Since steel has a greater capacity to hold heat than Styrofoam, more heat is pulled from your hand into the steel than is pulled into the Styrofoam. Hence the steel feels colder.
Note that since we are on Earth, not only is the temperature at -10 but the pressure is also constant at one atmosphere. This is a constant temperature and pressure situation.
The hotness and coldness we feel is a physiological response to more heat or less heat flowing out of you into the object.
Heat capacity, call it C, is a property of matter. The units are heat (Joules) per degree (Celsius, Kelvin or Farenheit) and the larger C, the more heat is required to raise its temperature.
Heat capacity is extensive (more substance, more heat it can hold). The main point here is that temperature is different from heat capacity. Temperature is a measure of the average energy of motion of a system (moving, vibrating and rotating mostly) while heat capacity is a measure of how much heat can flow across a boundary (say from your hand into something colder).
From this we can conclude that the heat capacity of steel is greater than the heat capacity of Styrofoam. We cannot know the heat capacity of our hands relative to the steel and Styrofoam because your hand is like a heat source and the steel will continue to pull heat from your body until you feel pain and remove your hand.
Of course be careful touching cold steel. Steel can pull heat from your hand so fast that your skin will freeze to it.
After this, it is possible to write down an equation which defines heat capacity. We have to specify constant pressure too, so
This is the mathematical way of expressing how much heat is required to change the temperature of a given amount of a substance; say one mole, at constant pressure. It says in words: “Heat capacity denotes the heat required, dq, to raise the temperature by dT at constant pressure. Of course the condition might not be constant pressure, for example it could be constant volume, so we define Cv.
Then do loads of examples.