The Equilibrium Constant
We have thus far discussed reactions which occur in the forward direction; creating products from reactants. However, these are not the only reactions that can occur. There are many types of reactions that can occur in both directions. These reactions are reversible and many help to maintain chemical equilibrium.
Definition – Reversible
A chemical reaction is reversible if it can proceed in the forward and reverse directions
Definition – Chemical Equilibrium
A reaction reaches chemical equilibrium when the rate of the forward reaction is equal to that of the reverse reaction.
These types of reactions are especially important in the human body, and other living organisms, to maintain homeostasis. However, can occur anywhere!
The equilibrium constant is a ratio between the products and the reactants to help determine “where” the reaction will settle at equilibrium.
Formula – Equilibrium Constant
For the balanced chemical reaction
$aA + bB ⇌ cC + dD$
The Equilibrium Constant ($K_c$) is written as:
$K_c=\frac{[C]^c[D]^d}{[A]^a[B]^b}$
Where [A] is the concentration (molarity) of A and,
a is the stoichiometric coefficient of A
It is important to note that purse solids (s) and pure liquids (l) are NOT to be included in the equilibrium constant calculation. Why not? Let’s look at liquid water, $\text{H}_2\text{O}(l)$. Recall that the density of liquid water is approximately 1 g/mL and the molar mass of water is 18.02 g/mol.
$1\frac{\text{g}}{\text{mL}}\cdot\frac{1\text{ mol}}{18.02\text{ g}}\cdot\frac{1\text{ mL}}{10^{-3}\text{ L}}$
$= 55.49 \frac{\text{mol}}{\text{L}}$
Thus the density of liquid water is 55.49 mol/L which is also the molarity of liquid water. And recall that density is an intensive property; it does not depend on the amount of substance.
Another important note is that the K is constant at a given temperature, despite the initial ratio of products to reactants. Only will a change in temperature change the value of K, due to Gibbs Free Energy.
Example – 1.1
Determine the equilibrium constant for the following balanced reaction
$\text{N}_2(\text{g})+3\text{H}_2(\text{g}) ⇋ 2\text{NH}_3(\text{g})$
Solution
Recall that $K_c=\frac{[C]^c[D]^d}{[A]^a[B]^b}$ where the numerator are the concentrations of the products raised to their stoichiometric coefficient and the denominator are those of the reactants
$\fbox{$K_c = \frac{[\text{NH}]^2}{[\text{N}_2][\text{H}_2]^3}$}$
So, what does K actually mean? Above we have $K_c$ which is the equilibrium constant for concentrations. There are other types of K such as $K_p$ for partial pressures, $K_a$ for acids, $K_b$ for bases. Each has the same formula structure but using the appropriate values for each respective type of K value.
$K_c$ is a ratio of the products to the reactants. If $K_c = 1$ this means the ratio of products to reactants is 1 at equilibrium
$K_c = \frac{[\text{products}]}{[\text{reactants}]} = 1$
The only way to get a ratio equal to 1 is if the numerator and denominator are equal to each other. Thus if $K_c=1$, then at equilibrium, there is an equal concentration of products to concentration of reactants.
Likewise, if $K_c > 1$, this means there is a larger concentration of products than reactants at equilibrium.
If $K_c < 1$, there is a larger concentration of reactants than products at equilibrium.
The equilibrium constant tells us WHERE the reaction will settle once it reaches equilibrium, NOT how fast it will approach equilibrium.
In Image 1, we can see that for K values greater than 1, the concentrations of the reactants starts out at 100% and limits to about 40% of the total concentrations at equilibrium. The concentration of the products starts at 0% and ends at around 60% of the total concentration.
In Image 2, the reactant concentration starts at 100% and ends at around 70% of the total concentration. Whereas the product concentrations start at 0% and end at around 30%. Thus giving us a K value less than 1.
Key Points – Equilibrium Constant
- If K << 1, the reverse reaction is favored (toward the reactants)
- If K $\approx$ 1, neither reaction is favored (at equilibrium)
- If K >> 1, the forward reaction is favored (toward the products)
- Solids and pure liquids are not included
- Aqueous dissolved particles are included
- Does not determine rate, only to what point at which equilibrium occurs
We can also find the equilibrium constant based on partial pressures, rather than concentrations. The formula will look very similar to that of $K_c$
Formula – Equilibrium Constant from Pressures
For the balanced chemical reaction
$aA + bB ⇌ cC + dD$
The Equilibrium Constant ($K_p$) is written as:
$K_p=\frac{P_C^cP_D^d}{P_A^aP_B^b}$
Where P_A is the partial pressure of A and,
a is the stoichiometric coefficient of A
This relationship comes from the ideal gas law PV=nRT, as pressures are proportional to concentrations.
Therefore, this relationship can derive the $K_p-K_c$ conversion formula
Formula – $K_c-K_p$ Conversion
$K_p=K_c(\text{RT})^{\Delta n}$
Where $\Delta n = n_{\text{products}}-n_{\text{reactants}}$, the difference between the number of moles of products to the number of moles of reactants and,
R is the universal gas constant $R = 0.08206\frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{k}}$ and,
T is temperature in Kelvin