Statistical Molecular Thermodynamics

Christopher J. Cramer

Video 10.1

Euler’s Theorem and Thermodynamics

A Little Formal MathematicsEuler’s Theorem

A homogeneous function of degree m exhibits the following behavior upon scaling of its arguments:

f λx1,λx2,!,λxn( ) = λm f x1, x2,!, xn( )

Proof proceeds by first differentiating both sides of eq. (1) by λ

(1)

Euler’s theorem asserts that for homogeneous functions:

mf x1, x2,!, xn( ) = xi∂f∂xii=1

n

∑ (2)

Proof of Euler’s Theorem

∂∂λ

f λx1,λx2,!,λxn( )"# $%=∂∂λ

λm f x1, x2,!, xn( )"# $%

∂f∂λxi

∂λxi∂λi=1

n

∑ =mλm−1 f x1, x2,!, xn( )

xi∂f∂λxii=1

n

∑ =mλm−1 f x1, x2,!, xn( )

xi∂f∂xii=1

n

∑ =mf x1, x2,!, xn( )

Homogeneity holds for all values of λ so we may set λ = 1, yielding

Q.E.D.

Relevance to Thermodynamics

Expressed in words, we might say, “double (or triple, or quadruple, etc.) your system size, and you double (or etc.) the value of your function.” And, that’s exactly how extensive thermodynamic properties, that are themselves functions of only extensive variables, behave!

f λx1,λx2,!,λxn( ) = λ f x1, x2,!, xn( )

Now, a homogeneous function of degree m = 1 implies

f λx1,λx2,!,λxn( ) = λm f x1, x2,!, xn( )

Relevance to Thermodynamics

Consider the particular example of Gibbs free energy at constant temperature and pressure for a two-component system, i.e.,

We may thus exploit Euler’s Theorem with m = 1

mf x1, x2,!, xn( ) = xi∂f∂xii=1

n

∑to determine:

f x1, x2,!, xn( ) = xi∂f∂xii=1

n

∑

G n1,n2( ) = n1∂G∂n1

+ n2∂G∂n2

Free Energy is the Sum ofChemical Potentials

G n1,n2( ) = n1∂G∂n1

+ n2∂G∂n2

We already have a definition for those derivatives, they are the chemical potentials (which do depend on P and T)

G n1,n2;P,T( ) = n1µ1 P,T( )+ n2µ2 P,T( )The notation emphasizes that we get to use Euler’s theorem for a fixed P and T, but the chemical potentials will themselves change at different values of P and T. A cleaner notation is:

G n1,n2( ) = n1µ1 + n2µ2

An Alternative Derivation

dG n1,n2,P,T( ) = ∂G∂n1

!

"#

$

%&P,T ,n2

dn1 +∂G∂n2

!

"#

$

%&P,n1,T

dn2 +∂G∂P!

"#

$

%&T ,n1,n2

dP + ∂G∂T!

"#

$

%&P,n1,n2

dT

From simply the total differential we already know:

which at constant P and T yields:

dG n1,n2;P,T( ) = µ1dn1 +µ2dn2

Considertion of appropriate boundary conditions and definite integration can then also yield

G n1,n2( ) = n1µ1 + n2µ2but Euler’s theorem is general for all extensive thermodynamic functions without the need to argue boundary conditions for each one separately

Next: Partial Molar Quantities and the Gibbs-Duhem Equation