Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 2.31 – 2.33.
The entropy of a substance is given as
where is the number of microstates accessible to the substance.
For a 3-d ideal gas, this is given by Schroeder’s equation 2.40:
where is the volume, is the energy, is the number of molecules, is the mass of a single molecule and is Planck’s constant. We can further approximate this formula by using Stirling’s approximation for the factorials:
When is large, we can throw away a couple of factors and take the logarithm:
This gives the entropy of an ideal gas as
which is known as the Sackur-Tetrode equation.
Example 1 A variant of this equation can be derived in a similar way for the 2-d ideal gas considered earlier. In that case we had
where is the area of the gas. Using Stirling’s approximation as before, we get
Example 2 Schroeder gives the entropy of a mole of helium at room temperature and atmospheric pressure as . For another monatomic gas such as argon, we can work out the same thing. From the ideal gas law, at a pressure of and temperature of 300 K, one mole occupies a volume of
The internal energy of a monatomic gas is per molecule per degree of freedom, so for one mole we have
The mass of a mole of argon is , so with we have
Plugging these values into 9, the entropy comes out to
This is a bit higher than the value for helium because of argon’s higher mass.