What are some of the equilibrium expressions that chemists use for weak acid equilibrium?

Ka = (products)/(reactants)

with coefficients becoming exponents.

Are there any more expressions for weak acids equilibrium? It would be helpful in the long run to know what they all are if there are any more.

You're question is so general that I didn't know what to write the first time and certainly I don't know what to write now. For a weak acid, such as HA, it ionizes as

HA(aq) + H2O(l) ==> H3O^+(aq) + A^-(aq) in which aq stands for aqueous (as in solution) and l stands for liquid (meaning pure H2O), the
Ka = (H3O^+)(A^-)/(HA)
All follow that same generic Ka. There are some diprotic and triprotic acids but they follow the same rules. For example, H2CO3 is
H2CO3(aq) + H2O(l) ==>H3O^+(aq) + HCO3^-(aq) and k1 for that ionization is
k1 = (H3O^+)(HCO3^-)/(H2CO3)

The second hydrogen then ionizes as
HCO3^-(aq) + H2O(l) ==>H3O^+(aq) + CO3^2-(aq) for which k2 is
k2 = (H3O^+)(CO3^2-)/(HCO3^-)
etc.

H3PO4, a triprotic acid, works the same way and it has a k1, a k2, and a k3 in which one H comes off at a time.
If you have a specific problem it would be easier to answer that than trying to stay with general cases.

Chemists use equilibrium expressions to describe the extent of a chemical reaction at equilibrium. For weak acid equilibria, there are a few commonly used equilibrium expressions, such as:

1. Acid dissociation constant (Ka):
The acid dissociation constant, Ka, describes the degree of dissociation of a weak acid in water. It is an expression of the equilibrium between the dissociated ions and the undissociated acid. The general equation for a weak acid dissociating in water is:
HA ⇌ H+ + A-
The equilibrium expression, Ka, can be written as:
Ka = [H+][A-] / [HA]
where [H+] represents the concentration of hydrogen ions, [A-] represents the concentration of the conjugate base, and [HA] represents the concentration of the weak acid.

2. Percent ionization:
Percent ionization is another expression used to describe the extent of ionization of a weak acid. It represents the percentage of the weak acid that ionizes in water. The equation for percent ionization is:
% ionization = (concentration of dissociated acid / initial concentration of acid) * 100

3. pH:
pH is a measure of the acidity or basicity of a solution and is closely related to the concentration of hydrogen ions (H+). For a weak acid equilibrium, the pH can be calculated using the following equation:
pH = -log[H+]
where [H+] represents the concentration of hydrogen ions.

To determine the actual values of these equilibrium expressions, you would typically need to know the initial concentration of the weak acid, as well as any values of the dissociation constants (Ka) or the pH of the solution.