Cloudscape

Below we come to a number of formulae which relate dimensionless heat capacity to heat in a striking way, showing, notably, that pressure times volume is a constant fraction of heat.

The formula of specific heat capacity is

c = Q/mT

According to the ideal gas law, Q = NkT on a microscopic level, so that

c = Nk/m = k/M

(where M is mass per molecule).

Q = cmT

Now, according to the ideal gas law,

PV = nRT
⇔ T = PV/nR

If we substitute this, we get:

Q = cmPV/nR

As m = nM,

Q = cMPV/R
⇔ PV = Q · (R/cM)

Now, heat capacity is the ability of a substance to store heat with increase of temperature. It is equal to the product of mass and specific heat capacity (the heated needed to heat a kilogram of a substance by one kelvin):

C = cm,

and, according to the heat equation,

Q = cmT,

so that this is equal to

C = Q/T

Now, the dimensionless heat capacity of a substance is equal to its heat capacity divided by the number of moles times the universal gas constant:

C* = C/nR,

the inverse of which is:

1/C* =nR/C

Now, the number of moles equals the total mass divided by the mass of one mole. Substituting m/M in n, we get:

1/C* =mR/MC,

and, as said, C = cm, so that

1/C* = R/cM.

Substituting this in the formula we got earlier, ideally:

PV = Q · (R/cM)

we get:

PV = Q/C*,

This formula may be useful because it relates heat, pressure and volume, showing that pressure times volume is a constant fraction of heat.
Now, according to kinetic theory,

P = ⅓ρv^2
⇔ PV = ⅓mv^2
⇔ PV = ⅔Ek

Thus,

⅔C*Ek = Q

This formula succinctly relates heat to kinetic energy. Note that kinetic energy equals work divided by two, so that

⅓C*W = Q

Also, according to the Dulong-Petit law, any crystal has a dimensionless heat capacity of 3, so that

2Ek = Q
⇔ W = Q

This means that, for any crystal, the heat of a crystal is the total work it may perform. This means that its internal energy twice its heat:

U = W+Q = 2Q = 2W