Homogeneous Functions, Euler's Theorem and Partial Molar Quantities

Any function f(x) that possesses the characteristic mapping:

\begin{array}{l} x \to λ x \\ f (x) \to λ f (x) \end{array}

(15.1)

is said to be homogeneous, with respect to x, to degree 1. By the same token, if f(x) obeys the mapping:

\begin{array}{l} x \to λ x \\ f (x) \to λ^{k} f (x) \end{array}

(15.2)

then f(x) is homogeneous to degree “k”. In general, a multivariable function f(x₁,x₂,x₃,…) is said to be homogeneous of degree “k” in variables x_i(i=1,2,3,…) if for any value of $λ$ ,

f (λ x_{1}, λ x_{2}, . . .) = λ^{k} f (x_{1}, x_{2}, . . .)

(15.3)

For example, let us consider the function:

f (x, y) = \frac{x}{x^{2} + y^{2}}

(15.4)

How do we find out if this particular function is homogeneous, and if it is, to what degree? We evaluate this function at $x = λ x a n d y = λ y$ to obtain:

f (λ x, λ y) = \frac{λ x}{λ^{2} x^{2} + λ^{2} y^{2}} = λ^{- 1} \frac{x}{x^{2} + y^{2}} λ^{- 1} f (x, y)

(15.5)

hence, the function f(x,y) in (15.4) is homogeneous to degree -1.

In regard to thermodynamics, extensive variables are homogeneous with degree “1” with respect to the number of moles of each component. They are, in fact, proportional to the mass of the system to the power of one (k=1 in equation 15.2 or 15.3). This is, if we triple the amount of mass in the system, the value of any given extensive property will be tripled as well. Notice that this is not the case for intensive properties of the system (such as temperature or pressure), simply because they are independent of mass. Hence, intensive thermodynamic properties are homogeneous functions with degree “0” — in such a case, k=0 in equation (15.2) or (15.3).

From the previous section, it is clear that we are not only interested in looking at thermodynamic functions alone, but that it is also very important to compute how thermodynamic functions change and how that change is mathematically related to their partial derivatives $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, and \frac{\partial f}{\partial z}$ . Hence, to complete the discussion on homogeneous functions, it is useful to study the mathematical theorem that establishes a relationship between a homogeneous function and its partial derivatives. This is Euler’s theorem.

Euler’s theorem states that if a function f(a_i, i = 1,2,…) is homogeneous to degree “k”, then such a function can be written in terms of its partial derivatives, as follows:

k λ^{k - 1} f (a_{i}) = \sum_{i}^{} a_{i} {(\frac{\partial f (a_{i})}{\partial (λ a_{i})}) |}_{λ x}

15.6a

Since (15.6a) is true for all values of $λ$ , it must be true for $λ - 1$ . In this case, (15.6a) takes a special form:

k f (a_{i}) = \sum_{i}^{} a_{i} {(\frac{\partial f (a_{i})}{\partial (a_{i})}) |}_{x}

15.6b

So far, so good. But…what is the application of all this? Well, first of all, we have to know something more about extensive thermodynamic properties. A very neat thing about them is that they can be written as a function of a sufficient number of independent variables to completely define the thermodynamic state of the system. Such a set is said to be a complete set. As it turns out, any thermodynamic system is completely defined when both the masses of all the substances within it are defined and two additional independent variables are fixed. This is Duhem’s theorem. From a real-life perspective, it is natural to choose pressure and temperature as those “independent variables” — physical quantities that we have a “feel” for and we think we can control — rather than specific volume or entropy. As we will see later, they are also convenient variables of choice because they are homogeneous of degree zero in mass.

Let “ $ℑ$ ” be a given extensive property of a multi-component system. From the previous section, we know that the value of “ $ℑ$ ” must be fixed and uniquely determined once we fix the pressure, temperature, and number of moles of each component in the system. This is,

ℑ = ℑ (P, T, n_{1}, n_{2}, . . ., n_{N})

15.7a

Additionally, we recall that extensive properties are homogeneous of degree one with respect to number of moles and homogeneous of degree zero with respect to pressure and temperature. Thus, expression (15.6b) is readily applicable:

ℑ = \sum_{i} n_{i} {(\frac{\partial ℑ}{\partial n_{i}})}_{P, T, n_{i = 1}} = \sum_{i} n_{i} ℑ_{i}

15.7b

where we have just defined:

ℑ_{i} {(\frac{\partial ℑ}{\partial n_{i}})}_{P, T, n_{i = 1}}

15.7c

Equation (15.7c) is a very important definition. It defines the concept of a partial molar quantity. A partial molar quantity $\bar{ℑ_{i}}$ represents the change in the total quantity ( $ℑ$ ) due to the addition of an infinitesimal amount of species “i” to the system at constant pressure and temperature. Euler’s theorem gave birth to the concept of partial molar quantity and provides the functional link between it (calculated for each component) and the total quantity. The selection of pressure and temperature in (15.7c) was not trivial. First, they are convenient variables to work with because we can measure them in the lab. But most important, they are intensive variables, homogeneous functions of degree zero in number of moles (and mass). This allowed us to use Euler’s theorem and jump to (15.7b), where only a summation with respect to number of moles survived. The definition of the partial molar quantity followed.

The conventional notation we are going to follow throughout the following section is:

$ℑ$ = Total quantity (e.g., total volume, total internal energy, etc),

$\tilde{ℑ}$ = Molar quantity, i.e., total quantity per unit mole:

\tilde{ℑ} = \frac{ℑ}{n} (for a mixture, n = \sum_{i} n_{i})

15.8a

$\bar{ℑ}$ = Partial molar quantity,

$\hat{ℑ}$ = Mass or specific quantity, i.e., total quantity per unit mass:

\hat{ℑ} = \frac{ℑ}{m} (for a mixture, m = \sum_{i} m_{i})

15.8b

We can rewrite equation (15.7b) in terms of molar quantity using the definition in (15.8a),

\tilde{ℑ} = \frac{1}{n} \sum_{i} n_{i} {\bar{ℑ}}_{i} = \sum_{i} x_{i} {\bar{ℑ}}_{i}

(15.9)

where:
$x_{i} = {molar fraction of species "i"= n}_{i} / n$

Any molar quantity in thermodynamics can be written in terms of the partial molar quantity of its constituents. If we set $ℑ = G$ , we end up with,

\tilde{G} = \sum_{i} x_{i} {\bar{G}}_{i}

(15.10a)

Equivalently, if we set $ℑ = V$ (total volume),

\tilde{v} = \sum_{i} x_{i} {\bar{v}}_{i}

(15.10b)

Notice that for single component systems (x_i=1), partial molar properties are just equal to the molar property:

\tilde{G} = {\bar{G}}_{i} {(if x}_{i} = 1, i = 1)

(15.11a)

\tilde{v} = v_{i} {(if x}_{i} = 1, i = 1)

(15.11b)

This is also a consequence of the definition in (15.7c),

{\bar{ℑ}}_{i} = {(\frac{\partial ℑ}{\partial n_{i}})}_{P, T, n_{i = 1}}

(15.7c)

For a pure component, $n = n_{i}, i=1, ℑ =n \tilde{ℑ}$ and:

\bar{ℑ} = {(\frac{\partial n \tilde{ℑ}}{\partial n})}_{P I} = \tilde{ℑ}

(15.12)

The reason for the introduction of the concept of a partial molar quantity is that often times we deal with mixtures rather than pure-component systems. The way to characterize the state of the mixtures is via partial molar properties. This concept provides the bridge between the thermodynamics of systems of constant composition, which we have studied so far, and the thermodynamics of systems of variable composition, which we will deal with in the next section. Basically, the definition in (15.7c):

{\bar{ℑ}}_{i} = {(\frac{\partial ℑ}{\partial n_{i}})}_{P, T, n_{i = 1}}

(15.7c)

allows us to quantify how the total, extensive property $ℑ$ changes with additions of n_i at constant pressure and temperature. If you look at (15.7b) and (15.9), you will also realize that (15.7c) is just an allocation formula that allows assigning to each species “i” a share of the total mixture property, such that:

ℑ = \sum_{i} n_{i} {\bar{ℑ}}_{i}

(15.7b)

\tilde{ℑ} = \sum_{i} x_{i} {\bar{ℑ}}_{i}

(15.9)

We can play with “ $ℑ$ ” a little more. Let us say that we are now interested in looking at the differential changes of $ℑ$ . Since we also know that $ℑ$ is a state function, and given the functional relationship in (15.7a), the total differential for $ℑ$ is written:

d ℑ = {(\frac{\partial ℑ}{\partial P})}_{T n_{i}} d P + {(\frac{\partial ℑ}{\partial T})}_{P, n_{i}} d T + \sum_{i} {(\frac{\partial ℑ}{\partial n_{i}})}_{P, T, n = 1} d n_{i}

(15.13a)

d ℑ = {(\frac{\partial ℑ}{\partial P})}_{T n_{i}} d P + {(\frac{\partial ℑ}{\partial T})}_{P, n_{i}} d T + \sum_{i} {\bar{ℑ}}_{i} d n_{i}

(15.13b)

Basically, equations (15.13) tell us that any change in P, T, or n_i will cause a corresponding change in the total property, $ℑ$ . This is a reinforcement of what is explicitly declared in (15.7a). If we recall (15.7b), an alternate expression for the total differential in (15.13) is written:

d ℑ = d (\sum_{i} n_{i} {\bar{ℑ}}_{i}) = \sum_{i} {\bar{ℑ}}_{i} d n_{i} + \sum_{i} n_{i} d {\bar{ℑ}}_{i}

(15.14)

If we subtract (15.14) from (15.13b), we get:

d ℑ - d ℑ = {(\frac{\partial ℑ}{\partial P})}_{T, n_{i}} d P + {(\frac{\partial ℑ}{\partial T})}_{P, n_{i}} d T + \sum_{i} {\bar{ℑ}}_{i} d n_{i} - \sum_{i} {\bar{ℑ}}_{i} d n_{i} - \sum_{i} n_{i} d {\bar{ℑ}}_{i}

(15.15)

Therefore,

{(\frac{\partial ℑ}{\partial P})}_{T, n_{i}} d P + {(\frac{\partial ℑ}{\partial T})}_{P, n_{i}} d T - \sum_{i} n_{i} d {\bar{ℑ}}_{i} = 0

(15.16)

Equation (15.16) is the well-known Gibbs-Duhem equation. It can be applied to any extensive thermodynamic property: U, S, H, G, A, and it must hold true. It represents a thermodynamic constraint between the intensive variables P, T and $\tilde{ℑ}$ . Pressure, temperature and partial molar properties cannot vary in just any fashion; any change taking place among them must satisfy (15.16). The change in any one of them can be calculated as a function of the change in the other two by means of the Gibbs-Duhem equation. This equation is the basis for thermodynamic consistency checks of experimental data.

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