Download The Euler Equation and the Gibbs-Duhem Equation and more Lecture notes Thermodynamics in PDF only on Docsity!
The Euler Equation and the Gibbs-Duhem
Equation
Stephen R. Addison
February 25, 2003
This document was included in the TEX-showcase. This paragraph was added by Gerben Wierda to display how TEX4ht displays the formatting of terms like LATEX and TEX.
1 Intensive Functions and Extensive Func-
tions
Thermodynamics Variables are either extensive or intensive. To illustrate the difference between these kings of variables, think of mass and density. The mass of an object depends on the amount of material in the object, the density does not. Mass is an extensive variable, density is an intensive variable. In thermodynamics, T , p, and μ are intensive, the other variables that we have met, U , S , V , N , H, F , and G are extensive. We can develop some useful formal relationships between thermodynamic variables by relat- ing these elementary properties of thermodynamic variables to the theory of homogeneous functions.
2 Homogeneous Polynomials and Homogeneous
Functions
A polynomial a 0 + a 1 x + a 2 x^2 + · · · + anxn
is of degree n if an 6 = 0. A polynomial in more than one variable is said to be homogeneous if all its terms are of the same degree, thus, the polynomial in two variables x^2 + 5xy + 13y^2
is homogeneous of degree two. We can extend this idea to functions, if for arbitrary λ
f (λx) = g(λ)f (x)
it can be shown that f (λx) = λnf (x)
a function for which this holds is said to be homogeneous of degree n in the variable x. For reasons that will soon become obvious λ is called the scal- ing function. Intensive functions are homogeneous of degree zero, extensive functions are homogeneous of degree one.
2.1 Homogeneous Functions and Entropy
Consider S = S(U, V, n),
this function is homogeneous of degree one in the variables U , V , and n, where n is the number of moles. Using the ideas developed above about homogeneous functions, it is obvious that we can write:
S(λU, λV, λn) = λ^1 S(U, V, n),
where λ is, as usual, arbitrary. We can gain some insight into the prop- erties of such functions by choosing a particular value for λ. In this case we will choose λ = (^1) n so that our equation becomes
S
( U
n
V
n
)
n
S(U, V, n)
Now, we can define Un = u, Vn = v and S(u, v, 1) = s(u, v), the internal energy, volume and entropy per mole respectively. Thus the equation becomes
ns(u, v) = S(U, V, n),
and the reason for the term scaling function becomes obvious.
and so for a j-component system
G =
∑^ j
i=
μini
4 The Gibbs-Duhem Equation
The energy form of the Euler equation
U = T S − pV + μn
expressed in differentials is
dU = d(T S) − d(pV ) + d(μn) = T dS + SdT − pdV − V dp + μdn + ndμ
but, we know that dU = T dS − pdV + μdn
and so we find 0 = SdT − V dp + ndμ.
This is the Gibbs-Duhem equation. It shows that three intensive variables are not independent – if we know two of them, the value of the third can be determined from the Gibbs-Duhem equation.
