**Heat capacity** is a term in physics that describes how much heat must be added to a substance to raise its temperature by 1 degree Celsius. This is related to, but distinct from, **specific heat**, which is the amount of heat needed to raise exactly 1 gram (or some other fixed unit of mass) of a substance by 1 degree Celsius. Deriving a substance's heat capacity C from its specific heat S is a matter of multiplying by the amount of the substance that is present and making sure you are using the same units of mass throughout the problem. Heat capacity, in plain terms, is an index of an object's ability to resist being warmed by the addition of heat energy.

Matter can exist as a solid, a liquid or a gas. In the instance of gases, heat capacity can depend on both ambient pressure and ambient temperature. Scientists often want to know the heat capacity of a gas at a constant pressure, while other variables such as temperature are allowed to change; this is known as the C_{p}. Similarly, it may be useful to determine a gas's heat capacity at a constant volume, or C_{v}. The ratio of C_{p} to C_{v} offers vital information about the thermodynamic properties of a gas.

## The Science of Thermodynamics

Before embarking on a discussion of heat capacity and specific heat, it is useful to first understand the basics of heat transfer in physics, and the concept of heat in general, and familiarize yourself with some of the fundamental equations of the discipline.

**Thermodynamics** is the branch of physics dealing with the work and energy of a system. Work, energy and heat all have the same units in physics despite having different meanings and applications. The SI (standard international) unit of heat is the joule. Work is defined as force multiplied by distance, so, with an eye on the SI units for each of these quantities, a joule is the same thing as a newton-meter. Other units you are likely to encounter for heat include the calorie (cal), British thermal units (btu) and the erg. (Note that the "calories" you see on food nutrition labels are actually kilocalories, "kilo-" being the Greek prefix denoting "one thousand"; thus, when you observe that, say, a 12-ounce can of soda includes 120 "calories," this is actually equal to 120,000 calories in formal physical terms.)

Gases behave differently from liquids and solids. Therefore, physicists in the world of aerodynamics and related disciplines, who are naturally very concerned with the behavior of air and other gases in their work with high-speed engines and flying machines, have special concerns about the heat capacity and other quantifiable physical parameters related to matter in this state. One example is **enthalpy**, which is a measure of the internal heat of a closed system. It is the sum of the energy of the system plus the product of its pressure and volume:

H = E + PV

More specifically, the change in enthalpy is related to the change in gas volume by the relationship:

∆H = E + P∆V

The Greek symbol ∆, or delta, means "change" or "difference" by convention in physics and math. In addition, you can verify that pressure times volume gives units of work; pressure is measured in newtons/m^{2}, while volume may be expressed in m^{3}.

Also, the pressure and volume of a gas are related by the equation:

P∆V = R∆T

where T is the temperature, and R is a constant that has a different value for each gas.

You don't need to commit these equations to memory, but they will be revisited in the discussion later about C_{p} and C_{v}.

## What Is Heat Capacity?

As noted, heat capacity and specific heat are related quantities. The first actually arises from the second. Specific heat is a state variable, meaning that it relates only to the intrinsic properties of a substance and not to how much of it is present. It is therefore expressed as heat per unit mass. Heat capacity, on the other hand, depends on how much of the substance in question is undergoing a heat transfer, and it is not a state variable.

All matter has a temperature associated with it. This may not be the first thing that comes to mind when you notice an object ("I wonder how warm that book is?"), but along the way, you may have learned that scientists have never managed to achieve a temperature of absolute zero under any conditions, though they have come agonizingly close. (The reason that people aim to do such a thing has to do with the extremely high conductivity properties of extremely cold materials; just think of the value of a physical electricity conductor with virtually no resistance.) Temperature is a measure of the motion of molecules. In solid materials, matter is arranged in a lattice or grid, and molecules are not free to move about. In a liquid, molecules are more free to move, but they are still constrained to a great extent. In a gas, molecules can move about very freely. In any event, just remember that low temperature implies little molecular movement.

When you want to move an object, including yourself, from one physical location to another, you must expend energy – or alternatively, do work – in order to do so. You have to get up and walk across a room, or you have to press the accelerator pedal of a car to force fuel through its engine and compel the car to move. Similarly, on a micro level, an input of energy into a system is required to make its molecules move. If this input of energy is sufficient to cause an increase in molecular motion, then based on the above discussion, this necessarily implies that the temperature of the substance increases as well.

Different common substances have widely varying values of specific heat. Among metals, for example, gold checks in at 0.129 J/g °C, meaning that 0.129 joules of heat is sufficient to raise the temperature of 1 gram of gold by 1 degree Celsius. Remember, this value does not change based on the amount of gold present, because the mass is already accounted for in the denominator of the specific heat units. Such is not the case for heat capacity, as you will soon discover.

## Heat Capacity: Simple Calculations

It surprises many students of introductory physics that the specific heat of water, 4.179, is considerably higher than that of common metals. (In this article, all values of specific heat are given in J/g °C.) Also, the heat capacity of ice, 2.03, is a less than half of that of water, even though both consist of H_{2}O. This shows that the state of a compound, and not just its molecular make-up, influences the value of its specific heat.

In any event, say you are asked to determine how much heat is required to raise the temperature of 150 g of iron (which has a specific heat, or S, of 0.450) by 5 C. How would you go about this?

The calculation is very simple; multiply the specific heat S by the amount of the material and the change in temperature. Since S = 0.450 J/g °C, the amount of heat that needs to be added in J is (0.450)(g)(∆T) = (0.450)(150)(5) = 337.5 J. Another way to express this is to say that the heat capacity of 150 g of iron is 67.5 J, which is nothing more than the specific heat S multiplied by the mass of the substance present. Obviously, even though the heat capacity of liquid water is constant at a given temperature, it would take vastly more heat to warm one of the Great Lakes by even a tenth of a degree than it would take to warm a pint of water by 1 degree, or 10 or even 50.

## What Is the Cp to Cv Ratio γ?

In a previous section, you were introduced to the idea of contingent heat capacities for gases – that is, heat-capacity values that apply to a given substance under conditions in which either the temperature (T) or the pressure (P) is held constant throughout the problem. You were also given the basic equations ∆H = E + P∆V and P∆V = R∆T.

You can see from the latter two equations that another way to express change in enthalpy, ∆H, is:

E + R∆T

Although no derivation is provided here, one way to express the first law of thermodynamics, which applies to closed systems and which you may have heard colloquially stated as "Energy is neither created nor destroyed," is:

∆E = C_{v}∆T

In plain language, this means that when a certain amount of energy is added to a system including a gas, and the volume of that gas is not allowed to change (indicated by the subscript V in C_{v}), its temperature must rise in direct proportion to the value of the heat capacity of that gas.

Another relationship exists among these variables that allows for the derivation of heat capacity at constant pressure, C_{p,} rather than constant volume. This relationship is another way of describing enthalpy:

∆H = C_{p}∆T

If you are adroit at algebra, you can arrive at a critical relationship between C_{v} and _{} C_{p}:

C_{p} = C_{v} + R

That is, the heat capacity of a gas at constant pressure is greater than its heat capacity at constant volume by some constant R that is related to the specific properties of the gas under scrutiny. This makes intuitive sense; if you imagine a gas being allowed to expand in response to increasing internal pressure, you can probably perceive that it will have to warm up less in response to a given addition of energy than if it were confined to the same space.

Finally, you can use all of this information to define another substance-specific variable, γ, which is the ratio of C_{p} to C_{v}, or C_{p}/C_{v}. You can see from the previous equation that this ratio increases for gases with higher values of R.

## The Cp and Cv of Air

The C_{p} and C_{v} of air are both important in the study of fluid dynamics because air (consisting of a mixture of mostly nitrogen and oxygen) is the most common gas that humans experience. Both C_{p} and C_{v} are temperature-dependent, and not precisely to the same extent; as it happens, C_{v} rises slightly faster with increasing temperature. This means that the "constant" γ is not in fact constant, but it is surprisingly close across a range of likely temperatures. For example, at 300 degrees Kelvin, or K (equal to 27 C), the value of γ is 1.400; at a temperature of 400 K, which is 127 C and considerably above the boiling point of water, the value of γ is 1.395.