How To Calculate Heat Capacity

If you watch the surface of a frozen pond slowly melt on an atypically warm winter afternoon, and watch the same thing occur on the surface of a nearby good-sized frozen puddle, you might observe the ice in each appear to be transformed to water at about the same rate.

But what if all of the sunlight falling on the exposed surface of the pond, maybe an acre in size, was simultaneously focused on the surface of the puddle?

Your intuition probably tells you that not only would the surface of the puddle melt into water very quickly, but the entire puddle might even become water vapor almost instantly, bypassing the liquid phase to become a watery gas. But why, from a physical science standpoint, should this be?

That same intuition is likely telling you that there is a relationship between heat, mass and the change in temperature of ice, water or both.

As it happens, this is the case and the idea extends to other substances as well, each of which have different "resistances" to heat, as manifested in different temperature changes in response to a given amount if added heat. These ideas combine to offer the concepts of specific heat and heat capacity.

Heat is one of the seemingly countless forms of the quantity known as energy in physics. Energy has units of force times distance, or newton-meters, but this is usually called the joule (J). In some applications, the calorie, equal to 4.18 J, is the standard unit; in still others, the btu, or British themal unit, rules the day.

Heat tends to "move" from warmer to cooler areas, that is, into regions into which there is presently less heat. While heat cannot be held or seen, changes in its magnitude can be measured via changes in temperature.

Temperature is a measure of the average kinetic energy of a set of molecules, like a beaker of water or a container of a gas. Adding heat raises this molecular kinetic energy, and hence the temperature, while reducing it lowers the temperature.

Why is a joule equal to 4.18 calories? Because the calorie (cal), while not the SI unit of heat, is derived from metric units and is fundamental in a way: It is the amount of heat needed to raise one gram of water at room temperature by 1 K or 1 °C. (A 1-degree change on the Kelvin scale is identical to a 1-degree change on the Celsius scale; however, the two are offset by about 273 degrees, such that 0 K = 273.15 °C.)

• The "calorie" on food labels is actually a kilocalorie (kcal) meaning that a 12-ounce can of sugary soda contains about 150,000 true calories.

The way one can determine such a thing through experimentation, using water or some other substance, is to place a given mass of it in a container, add a given amount of heat without allowing any of the substance or heat to escape the assembly, and measure the change in temperature.

Since you know the mass of the substance, and can assume that heat and temperature are uniform throughout, you can determine by simple division how much heat would change a unit amount, like 1 gram, by the same temperature.

The heat capacity formula comes in various forms, but they all amount to the same basic equation:

Q = mCΔT

This equation simply states that the change in heat Q of a closed system (a liquid, gas or solid material) is equal to the mass m of the sample times the temperature change ΔT times a parameter C called specific heat capacity, or just specific heat. The higher the value of C, the more heat a system can absorb while maintaining the same temperature increase.

Heat capacity is the amount of heat needed to increase the temperature of an object by a certain amount (usually 1 K), so the SI units are J/K. The object may be uniform, or it may not be. It would be possible to roughly determine the heat capacity of a mixture of substances such as mud if you knew its mass and measured its temperature change in response to heating it in a sealed device of some sort.

A more useful quantity in chemistry, physics and engineering is specific heat capacity C, measured in units of heat per unit mass. Specific heat capacity units are usually joules per gram-kelvin, or J/g⋅K, even though the kilogram (kg) is the SI unit of mass. One reason specific heat is useful is that if you have a known mass of a uniform substance and know its heat capacity, you can judge its fitness to serve as a "heat sink" to avoid fire risks in certain experimental situations.

Water actually has a very high heat capacity. Considering that the human body must be able to tolerate the addition or subtraction of significant amounts of heat thanks to Earth's varying conditions, this would be a basic requirement of any biological entity that is made mostly of water, as almost all sizable living things are.

Imagine a sports stadium that seats 100,000 people, and another across town that seats 50,000 people. At a glance, it is clear that the absolute "seat capacity" of the first stadium is twice that of the second. But also imagine that the second stadium is constructed in such a way that it takes up only one-fourth of the volume of the first.

If you do the algebra, you find that the smaller stadium actually seats twice as many people per unit space as the larger one, giving it twice the "specific seat" value.

In this analogy, think of individual spectators as units of heat of identical magnitude, flowing into and out of the stadium. While the larger stadium can hold twice as much "heat" overall, the smaller stadium actually has twice the capacity to "store" this version of "heat" per unit space.

If each same-sized section of both stadium is assumed to produce the same amount of post-game trash when full, regardless of how many people it holds, then the smaller one will be twice as effective at reducing the litter of individual spectators; think of this as being twice as resilient to temperature increases per unit of heat added.

From this, you can see that if two objects with the same specific heat have different masses, the larger one will have a larger heat capacity by an amount that scales with how much more massive it is. When comparing objects of different masses and different specific heats, the situation becomes more complex.

The metal copper has a specific heat of 0.386 J/g⋅K. How much heat is needed to raise the temperature of 1 kg (1,000 g, or 2.2 pounds) of copper from 0 °C to 100 °C?

Q = (m)(C)(ΔT) = (1,000 g)(0.386 J/g⋅K)(100 K) = 38,600 J = 38.6 kJ.

What is the heat capacity of this chunk of copper? You need 38,600 J to raise the entire mass by 100 K, so you would need 1/100th of this to nudge it up by 1 K. Thus the heat capacity of copper in this size is 386 J.