The change in free energy accompanied by isothermal reversible expansion of 1 mole of an ideal gas when it doubles its vol. Is delta G1. The change in free energy accompanied by sudden isotherm irreversible expansion of 1 mole of the same gas is delta G2. Ratio of G1 and G2 is

To understand the changes in free energy during the isothermal reversible and irreversible expansions of an ideal gas, we need to delve into some thermodynamic principles. Let's break down the concepts and calculations involved to find the ratio of ΔG1 (the change in free energy for reversible expansion) and ΔG2 (the change in free energy for irreversible expansion).

Reversible Expansion of an Ideal Gas

For a reversible isothermal expansion of an ideal gas, the change in Gibbs free energy (ΔG) can be calculated using the formula:

• ΔG = ΔH - TΔS

In an isothermal process, the change in enthalpy (ΔH) is zero for an ideal gas, so the equation simplifies to:

• ΔG = -TΔS

Now, for an ideal gas expanding isothermally from volume V to 2V, the change in entropy (ΔS) can be calculated using:

• ΔS = nR ln(Vf/Vi)

Here, Vf is the final volume (2V), Vi is the initial volume (V), and n is the number of moles (1 mole in this case). Plugging in the values:

• ΔS = 1 * R * ln(2V/V) = R ln(2)

Substituting this back into the ΔG equation gives:

• ΔG1 = -T(R ln(2)) = -RT ln(2)

Irreversible Expansion of an Ideal Gas

For the irreversible expansion of the same gas, the change in Gibbs free energy can be calculated differently. In this case, the gas expands freely into a vacuum, and the change in free energy is given by:

• ΔG = -nRT ln(Pf/ Pi)

For an irreversible expansion into a vacuum, the final pressure (Pf) becomes zero, and the initial pressure (Pi) is the pressure of the gas before expansion. However, since we are considering the expansion to a volume that doubles, we can use the same entropy change as before:

• ΔG2 = -RT ln(2)

Calculating the Ratio of ΔG1 to ΔG2

Now that we have both ΔG1 and ΔG2, we can find the ratio:

• ΔG1 = -RT ln(2)
• ΔG2 = -RT ln(2)

Thus, the ratio of ΔG1 to ΔG2 is:

• Ratio = ΔG1 / ΔG2 = (-RT ln(2)) / (-RT ln(2)) = 1

Final Thoughts

The ratio of the change in free energy for the reversible expansion to that of the irreversible expansion of 1 mole of an ideal gas, when it doubles its volume, is 1. This indicates that, despite the different processes, the free energy changes are equivalent in this specific scenario. Understanding these concepts helps clarify the behavior of gases under different thermodynamic conditions and the significance of reversibility in thermodynamic processes.