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Giuseppe Mensitieri » 10.Multicomponent Systems. Part 1

Multicomponent Uniform Systems

We analyze now multicomponent systems and, for the sake of simplicity, we consider here uniform multicomponent systems. In such a case, the local state and the total state of our body can be identified as follows:
$\sigma _{t}\equiv \left \{ V,\, T,\,\underline{n} \right \}$ and $\{V, T, \underline y\}$

With $\underline{y}\equiv\:molar\:fraction\:vector$ . Assuming the invertibility of volume and pressure, for the total volume of the system, $V_{t}$ , we have:
$V_{t}=V_{t}\left ( p,\:T,\:n_{1},.....\:n_{c}\right)$

Hence, for an infinitesimal change in state we have:

$dV_{t}=\left [ \left ( \frac{\partial V_{t}}{\partial T} \right )_{p,\, n_{k}} \right ]dT+\left [ \left ( \frac{\partial V_{t}}{\partial p} \right )_{T,\, n_{k}} \right ]dp+\left [ \left ( \frac{\partial V_{t}}{\partial n_{1}} \right )_{p,\,T,\, n_{2},....,n_{c}} \right ]dn_{1}+.....+\left [ \left ( \frac{\partial V_{t}}{\partial n_{c}} \right )_{p,\,T,\, n_{1},....,n_{c-1}} \right ]dn_{c}$

or, in a more compact form:

$dV_{t}=\left [ \left ( \frac{\partial V_{t}}{\partial T} \right )_{p,\, n_{k}} \right ]dT+\left [ \left ( \frac{\partial V_{t}}{\partial p} \right )_{T,\, n_{k}} \right ]dp+\sum_{k=1}^{c}\left [ \left ( \frac{\partial V_{t}}{\partial n_{k}} \right )_{p,\,T,\, n_{j}\neq n_{k}} \right ]dn_{k}$

Partial Molar Properties

We give now the following definition of partial molar volume as:

$V'_{t,\, k}\equiv \left ( \frac{\partial V_{t}}{\partial n_{k}} \right )_{p,\,T,\, n_{j}\neq n_{k}}$

This definition can be extended to the general case of any extensive property (i.e. $U_{t},\:A_{t},\:G_{t},\:H_{t},\:S_{t}$ ). Hence, we have for a generic extensive property:

$dB_{t}=M\,dT+Ndp+\sum_{k=1}^{c}B'_{t,\, k}dn_{k}$ (1)

where, by definition, the partial molar property $B'_{t,\, k}$ is:
$B'_{t,\, k}\equiv \left ( \frac{\partial B_{t}}{\partial n_{k}} \right )_{p,\,T,\, n_{j}\neq n_{k}}$ (2)

We look now at the consequences of such definition. Consider a process at constant T and p, where our system is formed by adding n1 moles of component 1, ……, n_c moles of component c until reaching a final state consisting of a homogeneous mixture of all the components. At each step of this process we have that:

$dV_{t}=\sum_{k=1}^{c}V'_{t,\, k}dn_{k}$

Partial Molar Properties

If we add the differential amounts of the components at the same time, in such a way that the composition of the mixture is always the same during the process , we have:

$V_{t}=\sum_{k=1}^{c}\int_{0}^{n_{k}}V'_{t,\, k}dn_{k}=\sum_{k=1}^{c}V'_{t,\, k}\int_{0}^{n_{k}}dn_{k}\; \; \;\Rightarrow\; \; \;V_{t}= \sum_{k=1}^{c}V'_{t,\, k}n_{k}$

In general we than have that:

$B_{t}= \sum_{k=1}^{c}B'_{t,\, k}n_{k}$ (3)

From eq. (3) we conclude that the total value of an extensive property is the weighed sum of the partial molar properties of each component of the mixture. This result derives from the fact that extensive properties are homogeneous functions of 1^st degree of the number of moles (see the following note).

Partial Molar Properties

Note on derivation of eq. (3)
If we consider a generic thermodynamic property, $\Phi _{t}$ , in the hypothesis that there are no phase changes, we have:
$\Phi _{t}=\phi\left(p,\:T,\:n_{1},\:....,\:n_{c}\right)\:\:\:\Rightarrow \:\:\:\Phi_{t}=\phi \left( p,\:T,\:\underline{n}\right)$

We remind here that, by definition, the value of extensive properties depends upon the amount of mass which is considered (examples of extensive properties are $U_{t},\:A_{t},\:G_{t},\:H_{t},\:S_{t} ,\:V_{t}$ ) while the value of intensive properties does not (examples of intensive properties are p and T).
For any extensive property we have that:
$\phi \left( p,\:T,\:\alpha\underline{n}\right)=\alpha\phi \left( p,\:T,\:\underline{n}\right)\: \: \forall\; \alpha >0$ (4)

Equation (4) implies that $\phi$ is a homogeneous function of 1st degree of $\underline{n}$ . We analyse now the mathematical consequences of this result. In fact, if we expand in a Taylor series the left hand side of the previous equation around $\underline{n}$ we obtain:

$\alpha\phi\left(p,\:T,\:\underline{n}\right)\:=\phi\left(p,\:T,\:\underline{n}\right)\:+\frac{\partial\phi}{\partial \underline{n}}\cdot\left[\left (\alpha-1\right)\underline{n}\right]+O\left[\left(\alpha -1\right)^2 \right]$ (5)

Now, if we subtract from both sides of eq. (5) we obtain:

$(\alpha -1)\phi(p, T, \underline n)=\frac{\partial \phi}{\partial \underline n}\cdot [(\alpha -1)\underline n]+O[(\alpha -1)^2]$ (6)

Partial Molar Properties

Then, dividing by (a-1):

$\phi(p,T,\underline n)=\frac{\partial \phi}{\partial \underline n}\cdot {\underline n}+\frac{O\lfloor(\alpha-1)^2\rfloor}{\alpha-1}\;\;\;\;\text{(7)}$

By taking the limit for $\alpha\rightarrow 1$ of eq.(7) we finally obtain:

$\Phi_t=\frac{\partial \phi}{\partial \underline n}\cdot{\underline n}=\left[\left(\frac{\partial \Phi_t}{\partial \underline n}\right )_{p,T,n_j\neq n_3}\right ]\cdot {\underline n}=\underline{\Phi '}_{t}}\cdot{\underline n}$

For example, if $\Phi_t=U_t$ we have:

$U'_{t,3}=\left(\frac{\partial U_t}{\partial n_3}\right)_{p,T,n_j\neq n_3}$

and

$U_t=\underline{U'_t}\cdot{\underline n}=U'_{t,1}n_{1}+...+U'_{t,c}n_c$

Partial Molar Properties

We can take now the differential of eq. (3), obtaining:

$dB_t=\sum_{k=1}^c d(B'_{t,k}n_k)$

from which:

$dB_t=\sum_{k=1}^c(B'_{t,k}dn_k+n_kdB'_{t,k})=\sum_{k=1}^c B'_{t,k}dn_k+\sum_{k=1}^c n_k dB'_{t,k}$

But, from eq (1) we have that, at constant T and p:

$dB_t=\sum_{k=1}^cB'_{t,k}dn_k$

We, then, conclude that, at constant T and p:

$\sum_{k=1}^cn_k dB'_{t,k}=0\;\;\;\text{(8) Gibbs-Duhem relationship}$

This is the GIBBS-DUHEM relationship from which we realize that the partial molar properties are not all independent. In particular, for a binary system, the Gibbs-Duhem relationship provides the basis for computing the values of partial molar properties (p.m.p) for one component when values for the other components have been already determined (Gibbs-Duhem integration).

Partial Molar Properties

Note on derivation of eq. (8)
If we consider now two mixtures, at the same T and P, such that:

$Mixture\: 1:\: \underline n,\: \underline{{\Phi}^{'}_{t}},\: {\Phi}_{t}$

$Mixture\: 2:\: \underline n\:+ d\underline n,\, \underline{{\Phi}^{'}_{t}}+d\underline{{\Phi}^{'}_{t}},\: {\Phi}_{t}+d{\Phi}_{t}$

We can calculate $d{\Phi}_{t}$ as follows:

$d{\Phi}_{t}\: =\: \frac {\partial \phi}_{\partial\underline n}}\cdot d\underline n\: =\: \underline{{\Phi}^{'}_{t}}\cdot d\underline n$

By evaluating $d{\Phi}_{t}$ in such a way we have not used the property of ${\Phi}_{t}$ of being an homogeneous function of order 1 of the number of moles.
In fact, if we instead calculate $d{\Phi}_{t}$ starting from

${\Phi}_{t} \: =\: \frac {\partial \phi}_{\partial\underline n}}\cdot \underline n\:=\: \left[{\left(\frac {\partial {\Phi }_{t}}_{\partial\underline n}}\right)}_{p,\, T,\,{n}_{j}\neq{n}_{k} }\right]\cdot \underline n\: =\: \underline{{\Phi}^{'}_{t}}\cdot \underline n$

which has been derived by considering that {\Phi}_{t} is a homogeneous function of order 1, we obtain:

Partial Molar Properties

$d{\Phi}_{t} \: =\: \underline{\Phi}^{'}_{t}\cdot d\underline n + \underline n \cdot d{\underline{\Phi}^{'}_{t}}}$

By comparing (a) and (b), we conclude that:

$\underline n \cdot d{\underline{\Phi}^{'}_{t}}}\:=\: 0 \: \: or\: \:\sum_{k=1}^c {n}_{k}d{\Phi }^{'}_{t,k}\: =\: 0$

Which is the Gibbs-Duhem relationship. Eq. (c) is a property only of the homogeneous function of order 1 of the number of moles. It states that when we change a little the composition of a mixture, all the partial molar properties change a little as well, but the scalar product of the vector of starting composition by the vector of variation of partial molar properties is zero.

It is worth noting that partial molar properties are homogeneous functions of order zero of the number of moles. As an example, if we consider the chemical potential vector, which is the vector of partial molar Gibbs Free Energy, we have:

$\underline \mu\: =\: \underline \mu\left(P,\, T,\, \underline n \right)\: =\: \underline \mu\left(P,\, T,\, \alpha \underline n \right)\: =\: \underline \mu\left(P,\, T,\, \underline x \right)$

(which has been obtained by taking $\alpha\: =\: \frac {1}_{\sum_{k=1}^{c}{n}_{k}}$ )

The Mixing Process

Temperature, pressure , volume and, according to the 3^rd law, entropy all have absolute values in thermodynamics. In contrast, there is no universally valid state for a system for which any of the energy functions (i.e. U_t, G_t, H_t, A_t) have a zero value. In fact they are always evaluated with respect to some reference state. Problems involving these functions, in general, deal only with changes in these values for processes. We consider here, in particular, the mixing process.

REFERENCE STATE for the formation of a solution: imagine to have containers with the single components in the physical state which is stable at the temperature and pressure of the solution to be formed.
MIXING PROCESS: it is the change in state experienced by the system when appropriate amounts of pure components in their reference state are mixed forming a homogeneous solution brought to the same temperature and pressure as the initial state. It is the formation of a solution from pure components at constant T and p.

Partial Molar Properties

Focusing always on uniform systems, we define with the symbol $\hat B_k^0$ the molar property of a pure component in a uniform state (i.e. all the thermodynamic properties have the same value all over the space occupied by the body). ‘0′ stands for ‘reference state’. We have:

$B_t^0=\sum_{k=1}^c\hat B_k^0 n_k\rightarrow \Delta B^{MIX}=B_t^{SOL}-B_t^0=\sum_{k=1}^c B'_{t,k}n_k-\sum_{k=1}^c\hat B_k^0 n_k=\sum_{k=1}^c(B'_{t,k - \hat B_k^0 n_k})$

From which we conclude that:

$\Delta B^{MIX}_t=\sum_{k=1}^c(B'_{t,k}-\hat B_k^0)n_k\;\;\;\text{(9)}$

By definition:

$\Delta B'_{t,k}\equiv B'_{t,k}-\hat B_k^0\;\;\;\text{(10)}$

it measures the change experienced by 1 mol of component k when it is transferred from its reference state to the surrounding environment it experiences in the solution at the selected composition. Consequently, form eq. (9) we obtain:

$\Delta B_t^{MIX}=\sum_{k=1}^c\Delta B'_{t,k}n_k\;\;\;\text{(11)}$

Partial Molar Properties

From this expression (11), it is evident that $\Delta B_t^{MIX}$ is the weighed sum of the changes experienced in the mixing process by the individual components. Now, from eq. (9), at constant T and p:

$d(\Delta B_t^{MIX})=\sum_{k=1}^c\left[B'_{t,k}dn_k+n_k dB'_{t,k}-\hat B_k^0 dn_k-n_k d\hat B_k^0\right]\;\;\;\text{(12)}$

By accounting for Gibbs-Duhem relationship and since ’s are independent of composition, expression (12) becomes:

$d(\Delta B_t^{MIX})=\sum_{k=1}^c(B'_{t,k}-\hat B_k^0 dn_{k})=\sum_{k=1}^c\Delta B'_{t,k}dn_k\;\;\;\;\text{(13)}$

Analogously, from eq. (11)

$d(\Delta B_t^{MIX})=\sum_{k=1}^c(\Delta B'_{t,k}dn_k+n_kd\Delta B'_{t,k})\;\;\;\;\text{(14)}$

By comparing (13) and (14) we finally obtain the form of the Gibbs-Duhem relationship applied to the mixing process

$\sum_{k=1}^cn_kd\Delta B'_{t,k}=0\;\;\;\text{(15)}$

Partial Molar Properties

At constant T and p we have:

Molar values of the properties of mixing can be easily obtained from the previous results by dividing the expressions by the total number of moles, i.e. n_t.