Mixing is a fascinating phenomenon in thermodynamics, particularly when considering the Gibbs energy of a mixture at constant temperature and pressure. This energy, denoted as G, tends to decrease during spontaneous mixing processes, offering insights into the composition changes that occur.
Imagine two ideal gases, initially separated in different containers, with amounts nA and nB, respectively, both at a temperature T and pressure p. The chemical potentials of these gases have their 'pure' values, determined by the equation μ = Gm, where Gm(p) = G°m + RT ln(p/p°).
Simplifying the notation by replacing p/p° with p, the total Gibbs energy of the separated gases is Gi. When these gases spontaneously mix, leading to partial pressures pA and pB, with pA+pB =p, the total Gibbs energy becomes Gf. The difference Gf − Gi represents the Gibbs energy of mixing, denoted as ΔGmix.
To express ΔGmix in terms of mole fractions (xJn), where n is the total amount of A and B and pJ/p = xJ for each component, gives the Gibbs energy of mixing for ideal gases. As mole fractions are never greater than 1, the logarithms in this equation are negative, confirming that ΔGmix<0. This implies that ideal gases mix spontaneously in all proportions.
Further exploration of the system reveals that (∂G/∂T)p = −S. It follows that, for a mixture of ideal gases at the same pressure, the entropy of mixing ( ΔSmix) is −(∂ ΔGmix/∂T)p. Since lnx<0, ΔSmix>0 for all compositions. The entropy of mixing increases for ideal gases at constant temperature and pressure, affirming their spontaneous mixing in all proportions.
Under constant pressure and temperature conditions, the enthalpy of mixing ( ΔHmix) can be calculated from ΔG = ΔH−TΔS. Notably, the enthalpy of mixing is zero, indicating no interactions between the molecules in the gaseous mixture. The driving force for mixing arises from the increase in system entropy, as the entropy of the surroundings remains unchanged.
In summary, the thermodynamics of mixing, as demonstrated through the Gibbs energy, entropy, and enthalpy changes, elucidates the spontaneous nature of ideal gas mixing in various proportions.
The thermodynamics of mixing describes the decrease in Gibbs energy during spontaneous mixing of ideal gases at constant temperature and pressure.
Consider two ideal gases with amounts nA and nB at the same temperature and pressure. Their chemical potentials follow from the molar Gibbs energy of an ideal gas.
The total Gibbs energy changes from Gi before mixing to Gf after mixing, with partial pressures pA and pB.
The Gibbs energy of mixing is obtained from Gf minus Gi and simplified using the relation between partial pressure and mole fraction of any gas J.
For an ideal gas, the partial derivative of G with respect to T at constant p equals minus the entropy. Applying this to the Gibbs energy of mixing gives the entropy change of mixing.
Since mole fractions are less than one, the logarithmic terms are negative, so mixing is spontaneous, with negative Gibbs energy of mixing and positive entropy of mixing.
From ΔG = ΔH − TΔS, the enthalpy of mixing is zero. Because unlike and like interactions are similar, mixing does not change enthalpy.
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