4.1
The Thermodynamics of Mixing
Mixing is a fascinating phenomenon in thermodynamics, particularly when considering the Gibbs energy of a mixture at constant temperature and pressure…
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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Q1: Why do ideal gases mix spontaneously?
Ideal gases mix spontaneously because the Gibbs energy of mixing is negative at constant temperature and pressure. Since mole fractions are always less than one, the logarithmic terms in the Gibbs energy equation become negative, driving spontaneous mixing in all proportions. This occurs regardless of the initial amounts of each gas.
Q2: How does entropy change when ideal gases mix?
The entropy of mixing is positive for ideal gases at constant temperature and pressure. Since mole fractions are less than one, the logarithmic terms are negative, making the entropy change positive. This increase in entropy is the driving force for spontaneous mixing, as the enthalpy of mixing remains zero.
Q3: What is the relationship between Gibbs energy and partial pressure during mixing?
The Gibbs energy of mixing depends on the partial pressures of each gas component. Using the relationship between partial pressure and mole fraction, the Gibbs energy change can be expressed as ΔGmix = RT(nA ln xA + nB ln xB). Since mole fractions are less than one, this yields negative Gibbs energy, confirming spontaneous mixing occurs.
Q4: Why is the enthalpy of mixing zero for ideal gases?
The enthalpy of mixing is zero because unlike and like molecular interactions are similar in ideal gases. From ΔG = ΔH − TΔS, with negative ΔG and positive ΔS, the enthalpy term must equal zero. This means no energy is absorbed or released during mixing; entropy increase alone drives the spontaneous process.
Q5: How do chemical potentials determine the total Gibbs energy before and after mixing?
Before mixing, the total Gibbs energy equals the sum of chemical potentials of separated gases at their pure pressures. After mixing, each gas has a lower chemical potential due to reduced partial pressure. The difference between final and initial Gibbs energy gives the Gibbs energy of mixing, which is negative for ideal gases.
Q6: What role does temperature play in the spontaneity of ideal gas mixing?
Temperature affects the entropy contribution to Gibbs energy through ΔG = ΔH − TΔS. Since enthalpy of mixing is zero and entropy of mixing is positive, the TΔS term is always positive. This makes ΔG negative at any temperature, ensuring ideal gases mix spontaneously regardless of temperature changes at constant pressure.
Q7: How are mole fractions used to express the Gibbs energy of mixing?
Mole fractions replace partial pressures in the Gibbs energy equation using the relationship pJ/p = xJ for each component. This simplification yields ΔGmix = RT(nA ln xA + nB ln xB), where logarithms of mole fractions are negative. This expression reveals why thermodynamic properties of ideal solutions show spontaneous mixing in all proportions.