Graphing the Gibbs free energy of mixed gases

[grandpa2390]

Homework Statement

I am needing to graph the Gibbs free energy of mixed gases to determine the range when the gases will form an ideal mixture
The two gases have the same Gibbs free energy.

Homework Equations

$G = (1-X)G_A + XG_B$ for unmixed
$G = (1-X)G_A + (X)G_B + RT(x*ln(x) + (1-X)*ln(1-X))$ for mixed
$U = 6nTx_Ax_B$ x_A and x_B are molar fractions of the gases.

The Attempt at a Solution

so my attempt is that since the Gibbs of both gases equal the same, then $G_A = G_B$ so $G = G_{AB} + RT(x*ln(x) + (1-X)*ln(1-X))$

I am not sure how to graph that. I don't know what T is.
the second formula assumes that U does not change. But in this problem, it appears that U does. So I am thinking that my formula would be to add U in that equation:

$G = G_{AB} + 6nT(x)(1-x) + RT(x*ln(x) + (1-x)*ln(1-x))$

I don't know how to graph this because there are 3 independent variables still... ( T, x, and n) But I have to graph it somehow in order to estimate the range where the gases will form an good mixture.

 

[anathema]

Hello,

Thank you for your post. I can assist you with graphing the Gibbs free energy of mixed gases. First, let's clarify some things about the equations you have provided.

The first equation you have provided, $G = (1-X)G_A + XG_B$, is actually the equation for the Gibbs free energy of an unmixed system, where X is the molar fraction of gas B and (1-X) is the molar fraction of gas A.

The second equation, $G = (1-X)G_A + (X)G_B + RT(x*ln(x) + (1-X)*ln(1-X))$, is the equation for the Gibbs free energy of a mixed system, where X is the molar fraction of gas B and (1-X) is the molar fraction of gas A. The term RT(x*ln(x) + (1-X)*ln(1-X)) is known as the entropy of mixing, which takes into account the randomness of the molecules in a mixed system.

Now, for your attempt at a solution, there are a few things to clarify. First, the Gibbs free energy of each gas, G_A and G_B, will not necessarily be equal. They may be equal at certain temperatures and pressures, but in general, they will be different. Second, the term U in your equation is not needed, as it is already taken into account in the entropy of mixing term.

To graph the Gibbs free energy of mixed gases, we need to first choose a temperature and pressure at which we want to graph. Let's say we choose a temperature of 298 K and a pressure of 1 atm. Now, we can use the equation $G = (1-X)G_A + (X)G_B + RT(x*ln(x) + (1-X)*ln(1-X))$ to calculate the Gibbs free energy of the mixed system at different molar fractions of gas B. We can then plot these values on a graph, with the molar fraction of gas B on the x-axis and the Gibbs free energy on the y-axis.

The resulting graph will have a shape similar to a parabola, with the lowest point representing the minimum Gibbs free energy of the system. This is the point at which the gases will form an ideal mixture, as the Gibbs free energy is at its lowest possible value. The range in which the gases will form an ideal mixture can