Page 39 - Thermodynamics of Biochemical Reactions

2.8 Calculation of Thermodynamic Properties of a Monatomic Ideal Gas 33

2001) is given as a function of 7: P, and n by
2nmkT 3/2 kT
G = -nRTln (2.8-1)

where nz is the mass of the atom, k is the Boltzmann constant, and h is Planck's
constant. Equations 2.5-7 to 2.5-9 show that S, and p can be calculated by
taking partial derivatives of G with respect to 7: P, and n. Taking these partial
derivatives yields
S = nR (In [(T) 2nmkT 3/2 kT + ;)
(2.8-2)

nRT
I/= __ (2.8-3)
P
P= -RTh [(T) (2.8-4)
2nmkT 3/2 kT

Equation 2.8-2 is referred to as the Sackur-Tetrode equation. Since we have
expressions for these three properties, we can calculate the properties U, H, A,
and Cp:

H = (;) nRT (2.8-5)

U = (i) nRT (2.8-6)
A = -nRTln [(')
- nRT
nmkT 3/2 kT
(2.8-7)

C, = (g) nR (2.8-8)

Note that U corresponds with the translational kinetic energy in three directions.
Thus all the thermodynamic properties of an ideal monatomic gas can be
calculated from G( 7; P, n).
Equations 2.8-2 and 2.8-4 can be used to derive the expressions for the
standard molar entropies and standard molar Gibbs energies of a monatomic gas:

S,=Sz-Rln - (2.8-9)
(Fp)

(2.8- 10)

where Po is 1 bar and
s: = R (In [ 2nmkT 3/2 kT + ;) (2.8-1 1)

G: = p' = -RTln (2.8 - 12)

These are the properties of the monatomic gas at a pressure of 1 bar. It should
be pointed out that this standard molar Gibbs energy is not the A,G" of
thermodynamic tables because there the convention in thermodynamics is that
the standard formation properties of elements in their reference states are set
equal to zero at each temperature. However, the standard molar entropies of
monatomic gases without electronic excitation calculated using equation 2.8-1 1
are given in thermodynamic tables.