Equilibrium constant

An equilibrium constant is a mathematical expression that specifies the relationship between reactant and product concentrations in a reversible chemical reaction. The equilibrium constant is always a ratio of equilibrium concentrations, and it doesn’t depend on the absolute concentrations of anything. It does, however, depend on temperature.

Equilibrium constant definition

The formula for Keq is derived from the definition of an equilibrium constant.

The relationship between Kp and Kc is given by the van’t Hoff equation:

ΔG=ΔH−TΔS

where T is the temperature in degrees Kelvin, R is the universal gas constant, Δn is changing in moles of products minus change in moles of reactants, and Δf is the change in free energy

Effect of temperature on equilibrium constant 

Temperature affects the equilibrium constant because it alters the energies of the molecules involved in the reaction. If you measure standard free energy at two different temperatures, then you’ll see that it has a different numerical value at each temperature, so there’s no way to equate these values mathematically. The key is to consider free energy in terms of entropy and enthalpy.

The standard change in free energy for a chemical reaction under any conditions is the difference between the sum of the products’ standard enthalpies – usually at 298 K – minus the sum of the reactants’ standard enthalpies. If you plot this value against reaction progress (Δn, the change in moles of products minus the change in moles of reactants at any given time), then you get a straight line with a slope equal to -R/2.

This means that Δf varies linearly with reaction progress. When you plot the natural logarithm of Kp against reaction progress on a graph, then it makes a straight line with a slope equal to the negative inverse of the slope of Δf versus reaction progress.

The formula for equilibrium constant is derived from a straight-line relationship between ln(Kp) and reaction progress. The negative inverse slope arises from the natural logarithm.

The formula for T is derived from a straight-line relationship between ln(Kp) and reaction progress when Δf = -R/2.

The activity coefficient of the equilibrium constant

The activity coefficient of the equilibrium constant is the relative concentration of any species to species present in water. If you take one of the constants on the right-hand side of this equation and raise it to the power of T/nR, then that’ll give you an approximation for temperature dependence. Since activity coefficients often don’t exceed 0.9 under standard conditions, if activity coefficient > 0.9 (and assuming X > 0), then at lower temperatures, activity coefficient approaches 1.

K=adD⋅aeE/abB⋅acC

Calculating equilibrium constant

This relationship approximates for activity coefficients exceeding 0.9 under standard conditions. It also approximates at lower temperatures when X > 0.

bB+cC⇌dD+eE

K=adD⋅aeE/abB⋅acC

This equation says that if T is low enough, then the concentration of water molecules becomes negligible compared to the concentration of other molecules in the solution. There’s no need to include water molecules when you calculate Q at low temperatures because their concentrations are essentially constant under standard conditions (10 M). This means that if X > 0, then as T approaches 0 K, equation 1 becomes:

where the activity coefficient of species X is in water at standard temperature and pressure. This equation says that if X > 0, then the equilibrium constant of a reaction approaches infinity as T approaches 0 K.

The formula for Q is derived from the definition of an equilibrium constant with no consideration of water’s concentration when X > 0 at low temperatures.

The result derived from equilibrium constant

As you can see, this result depends on the activity coefficients of both reactants and products in water. This is because the concentration of water is constant under standard conditions (10 M), but the concentrations of other molecules aren’t when you’re at low temperatures. This means that when X > 0, even a small decrease in twill causes a significant increase in Q.

The formula for Kp is derived from the definition of a Kp value. This equation gives you an approximate value for Kp at high temperatures as the mass action law doesn’t apply under those conditions.

This equation says that as T approaches 0, then Δn approaches zero and will be equal to m initially if no other changes occur. Since Δn is equal to the negative of m, this means that Δn approaches -m.

Calculating equilibrium constant from mass action law

The formula for Kp is derived from the mass-action law applied to a constant concentration of reactants and products under high temperatures. This equation is still valid when you consider that Δn approaches -m at high temperatures.

This equation shows how changing T affects the equilibrium constant, given that Δn = -m. Since Δn is equal to the negative of m, this means that as T increases, Kp decreases and becomes more negative (indicating a smaller amount of products compared to reactants).

The formula for Kp is derived from the definition of an equilibrium constant at high temperatures.

Conclusion

As you can see, this result is similar to ΔG=ΔH−TΔS.The only difference is that instead of using the activity coefficients γ (which are different for every species in an equilibrium), We use the steam table average value for all substances at standard temperature and pressure. This means that as T approaches 0 K, then Δn becomes zero, and Kp approaches -nRT/Z, which is the same as -n.

The formula for Kp is derived from the definition of an equilibrium constant with no consideration of species’ activity coefficients when T increases. This equation approximates at high temperatures.