# Specific heat and heat capacity

## Definitions

Specific heat is a measure of how much thernal energy is required for one unit of mass of a substance in order to change its temperature by one unit of temperature.

The heat capacity of a mass is a measure of how much thermal energy a mass can release (or absorb) by a change of one unit temperature. If the mass is homogenous, its heat capacity is equal to its specific heat times its total mass.

$C = c m$

where:
C is the heat capacity
c is the specific heat
m is the mass

A related quantity, the heat capacity rate, is a measure of the amount of heat a flowing fluid can release or absorb per unit time, given a unit change in temperature.

$C = {dC \over dt} = c {dm \over dt}$

where:
${dC \over dt}$ is the heat capacity rate
${dm \over dt}$ is the mass flow rate

## Calculating Specific Heat

Since a substance’s instantaneous specific heat varies with temperature, the total thermal energy of a change in temperature $\Delta$T must be derived by integrating the heat capacity over the temperature change involved.

$dQ = C dT$

When dealing with solids and liquids, if the temperature change is small enough (up to 100K, depending on the substance), and the specific heat at the starting temperature is known, then assuming the specific heat constant can yield a reasonably accurate result.

$\Sigma Q = C_{const} \Delta T$

However, the specific heats of gases are not constant enough for this assumption to hold over a range of more than a few degrees.

Only the specific heats of monoatomic noble gases can be accurately derived from theory and remain relatively constant over broad temperature ranges.

$c = {5 R \over 2 W}$

where:
R is the universal gas constant (8.314 J/K-mol)
W is the molecular weight of the noble gas

Even these relatively predictable gases will vary in specific heat as they approach their boiling and ionization points.

The specific heats of all other gases must be determined empirically at the temperature involved. The only means of ensuring an accurate value of specific heat for a gas is to measure it. Once empirical data is in hand, however, empirical formulas can be fitted to that data to project a gas’s specific heat at a given temperature. Over a large range (up to 8000K, depending on the substance), use of empirical formulas can give highly accurate values for the specific heat at a given temperature. Almost all complex molecules undergo state transitions at sufficientl;y high temperatures, and it is the positions of these state transitions in the temperature range, rather than the size of the range used, that determines the accuracy of an empirical formula for specific heat.

Given the wide temperature variations expected on the Martian surface, variations in specific heat should be considered for all design calculations.

## Specific Heat of a Mixture

A mixture of two substances, for which the specific heats are known, will have a specific heat of:

$c_{mix} = {C_{1} + C_{2} \over m_{1} + m_{2}}$

For example, the Martian atmosphere is roughly 95% carbon dioxide and 3% nitrogen and 2% argon. At –30oC, the specific heat of nitrogen is 1042 J/kgK, argon’s is 522.2 j/kgK and that of carbon dioxide is 804 J/kgK Thus, by the formula above, the specific heat of the Martian atmosphere at that temperature is roughly 805 J/kgK, or nearly the same as that of carbon dioxide, its major constituent.

If the heat capacity of one substance is much less than that of the other, the mixture’s specific heat can be assumed equal to that of the more abundant substance.