A container is partitioned into two compartments, one of which contains 2 moles of He while other contains 3 moles of At.The gases are ideal. The temperature is 300K and the pressure is 1 bar.Calculate total Gibbs free energy of two gases?

To calculate the total Gibbs free energy of the two gases in the container, we can use the formula for the Gibbs free energy of an ideal gas. The Gibbs free energy (G) for an ideal gas can be expressed as:

Gibbs Free Energy Formula

The formula for the Gibbs free energy of an ideal gas is given by:

G = nRT ln(P) + nG₀

Where:

• G = Gibbs free energy
• n = number of moles of the gas
• R = universal gas constant (approximately 8.314 J/(mol·K))
• T = temperature in Kelvin
• P = pressure in bar
• G₀ = standard Gibbs free energy of formation (which we will assume to be zero for simplicity)

Calculating Gibbs Free Energy for Helium (He)

For helium, we have:

• n_He = 2 moles
• T = 300 K
• P = 1 bar

Substituting these values into the formula:

G_He = 2 moles × 8.314 J/(mol·K) × 300 K × ln(1 bar)

Since ln(1) = 0, we find:

G_He = 0 J

Calculating Gibbs Free Energy for Astatine (At)

Now, for astatine, we have:

• n_At = 3 moles
• T = 300 K
• P = 1 bar

Using the same formula:

G_At = 3 moles × 8.314 J/(mol·K) × 300 K × ln(1 bar)

Again, since ln(1) = 0, we find:

G_At = 0 J

Calculating Total Gibbs Free Energy

Now, to find the total Gibbs free energy of the system, we simply add the Gibbs free energies of both gases:

Total G = G_He + G_At = 0 J + 0 J = 0 J

Final Thoughts

In this scenario, both gases are at the same pressure and temperature, and since the natural logarithm of 1 is zero, the Gibbs free energy for both gases turns out to be zero. Therefore, the total Gibbs free energy of the system is also zero. This result emphasizes the concept that at equilibrium conditions (like the ones given), the Gibbs free energy change is zero, indicating no net change in the system's energy state.