User:Dbrogioli/Calculation of consumed free energy

Calculation of osmotic pressure from Gibbs free energy

Take a volume V of salt water at concentration c; put an osmotic membrane; put a small volume v of fresh water on the other side. The osmotic pressure is $\Pi \left(c\right)$. When the volume v passes into the more concentrated solution, a work is done; in isothermal conditions, it is equal to the Gibbs free energy decrease:

$\Pi \left(c\right)v=Vg\left(c\right)-\left(V+v\right)g\left({\frac {cV}{V+v}}\right)$

where $g\left(c\right)$ is the Gibbs free energy density (i.e. per unit volume of solution).

By taking the limit $v/V\to 0$:

$\Pi \left(c\right)=g\left(c\right)-c{\frac {\partial }{\partial c}}g\left(c\right)$

Calculation of the Gibbs free energy density

Once $\Pi \left(c\right)$ is known, $g\left(c\right)$ can be calculated, but is defined up to a constant $\delta$ and a linear term $\gamma c$: any term $\delta +\gamma c$ can be added without changing the equality.

Any linear term is negligible when considering free energy changes in mixing (see below).

This allows to define the Gibbs free energy density $g\left(c,c_{0}\right)$ relative to a concentration $c_{0}$, that is the Gibbs free energy density up to a constant and linear term $\delta +\gamma c$, which moreover has the property that:

$g\left(c,c_{0}\right)=0$

and

${\frac {\partial }{\partial c}}g\left(c,c_{0}\right)=0$

Free energy changes in mixing

Consider a Gibbs free energy density $g\left(c\right)$ defined up to a constant and linear term $\delta +\gamma c$. Consider two volumes $V_{A}$ and $V_{B}$ of solution at concentrations $c_{A}$ and $c_{B}$. The total free energy is:

$F_{initial}=V_{A}g\left(c_{A}\right)+\delta V_{A}+\gamma V_{A}c_{A}+V_{B}g\left(c_{B}\right)+\delta V_{B}+\gamma V_{B}c_{B}$

Now, we mix the two solutions. For NaCl solutions, the volumes are summed:

$V_{total}=V_{A}+V_{B}$

while the total concentration is:

$c_{total}={\frac {c_{A}V_{A}+c_{B}V_{B}}{V_{A}+V_{B}}}$

The total free energy becomes:

$F_{final}=\left(V_{A}+V_{B}\right)g\left({\frac {c_{A}V_{A}+c_{B}V_{B}}{V_{A}+V_{B}}}\right)+\delta \left(V_{A}+V_{B}\right)\gamma \left(V_{A}+V_{B}\right){\frac {c_{A}V_{A}+c_{B}V_{B}}{V_{A}+V_{B}}}$

We immediately see that the difference $\Delta F=F_{initial}-F_{final}$ does not depend on $\delta$ and $\gamma$:

$\Delta F=V_{A}g\left(c_{A}\right)+V_{B}g\left(c_{B}\right)-\left(V_{A}+V_{B}\right)g\left({\frac {c_{A}V_{A}+c_{B}V_{B}}{V_{A}+V_{B}}}\right)$

The same calculations apply also to the Gibbs free energy density $g\left(c,c_{0}\right)$ relative to a concentration $c_{0}$; as we have seen, we can use any value of $c_{0}$ for calculations of free energy changes, since $g\left(c,c_{0}\right)$ differs from $g\left(c\right)$ only by a constant and linear term.

Free energy change when water is dispersed into the sea

This is a particular case of the previous section. In particular, $V_{A}=v$ is a small quantity with respect to the volume of the sea $V_{B}$. Taking the limit:

$\Delta F=v\left[g\left(c_{A}\right)-g\left(c_{B}\right)-\left(c_{A}-c_{B}\right){\frac {\partial }{\partial c}}g\left(c_{B}\right)\right]$

It is useful to use the Gibbs free energy density $g\left(c,c_{0}\right)$ relative to a concentration $c_{B}$. In this case, the equation simplifies to:

$\Delta F=vg\left(c_{A},c_{0}\right)$

When we chose a value $c_{0}$, the function $g\left(c,c_{0}\right)$ defines the free energy density with respect to the solution disposed into a sea with concentration $c_{0}$.