# 26.2: An Equilibrium Constant is a Function of Temperature Only

### Equilibrium constants

If the reaction goes to equilibrium ΔrG goes to zero. In that case Q is usually rewritten as the equilibrium constant $$K$$ and we get:

0= ΔrG = ΔrGo + RTlnK
ΔrGo = - RTlnK

where K=[Peq,YvYPeq,ZvZ /Peq,AvAPeq,BvB]

Note

As you see $$Δ_rG^o$$ is not zero, because the standard state does not represent an equilibrium state (typically).

Again the values of both K and ΔrGo depend on what the standardo was chosen to be.

### Le Chatelier

In the ideal gas reaction case K only depends on T (just like U) not on the total pressure. This leads to the well-known principle of Le Chatelier. Consider a gas reaction like:

$A \rightleftharpoons B + C$

e.g.

$PCl_5 \rightleftharpoons PCl_3 + Cl_2$

In pressures, the equilibrium constant becomes:

$K = \dfrac{P_BP_C}{P_A}$

If initially nA =1 we have at an extent ξ:

nA = 1- ξ
nB= ξ
nC= ξ

nTotal = 1+ ξ

The partial pressures are given by Dalton's law:

PA= [1- ξ /1+ ξ] P
PB [ξ / 1+ ξ ] P
PC [ξ /1+ ξ ] P

The equilibrium constant becomes:

$K = P\dfrac{ξ_{eq}^2}{1-ξ_{eq}^2}$

Even though the total pressure P does occur in this equation, K is not dependent on P. If the total pressure is changed (e.g. by compression of the gas) the value of ξeq will change (the equilibrium shifts) in response. It will go to the side with the fewer molecules. This fact is known as le Chatelier's principle

If the system is not ideal we will also get a Chatelier shift, but the value of K may change a little because the value of the activity coefficients (or fugacity) is a little dependent on the pressure too. In solution the same thing holds. In ideal solutions K is only T dependent, but as we saw these systems are rare. Particularly in ionic solutions equilibrium constants will be effected by other things than just temperature, e.g. changes in the ionic strength and we need to find the activity coefficients to make any predictions.

### Concentration

The gas law contains a hidden definition of concentration:

$PV= nRT$

$P= \left(\dfrac{n}{V} \right) RT$

$P= c RT$

$c= \dfrac{P}{RT} \label{Conc}$

Here c stands for the molar amount per unit volume or molarity. For gaseous mixtures we do not use this fact much, but it provides the link to the more important liquid solution as a reaction medium. We can rewrite the equilibrium constant as

$K= \dfrac{c_{eq,Y}^{v_Y} c_{eq,Z}^{v_Z} }{c_{eq,A}^{v_A} c_{eq,B}^{v_B}} \label{K}$

However, c (Equation $$\ref{Conc}$$) is substituted into $$K$$ (Equation $$\ref{K}$$), then not all the factors of $$RT$$ cancel. The missing term g. ln[RT] depends on the stoichiometric coefficients:

$g=v_Y+v_Z-v_A-v_B$

The term is generally incorporated in ΔrGo so that the latter now refers to a new standard state of 1 mole per liter of each species rather than 1 bar of each (or so).