pH calculations

In a nutshell

The pH scale indicates if the solution is acidic, neutral or basic. It measures the concentration of hydrogen ions. There are calculations that can be used to find the pH of strong and weak acids.

Equations

Variable units

pH scale and strong acids

The pH scale indicates if the solution is acidic, neutral or basic. It measures the concentration of hydrogen ions, also called protons. When a solution has a high concentration of hydrogen ions, it is likely to be acidic. When a solution has a low concentration of hydrogen ions, it is likely to be basic. If the concentration of hydrogen ions is equal to the concentration of hydroxide ions, the solution will be neutral. 

The pH scale is measured in numbers up to $$14$$​. Anything lower than seven is acidic, seven is neutral and anything above seven is basic. The pH is calculated from the concentration of hydrogen ions using this equation.

​$$pH=-\log _{10}[H^+]$$

Strong acids dissociate fully in a solution. This means the concentration of the acid is equal to the concentration of hydrogen ions present in a solution. 

Example

Calculate the pH of $$0.031\>mol\>dm^{-3}$$ of hydrochloric acid. 

Write down the pH equation.

$$pH=-\log _{10}[H^+]$$

Hydrochloric acid is a strong acid so the concentration of $$H^+$$ will be the same as the acid.​​

$$[H^+]=[HCl]=0.031\>mol\>dm^{-3}$$

Insert the concentration of $$H^+$$in the pH equation to calculate the pH.

$$pH=-\log _{10}[0.031]=1.51$$

The pH is $$\underline{1.51}$$.

Note: Always give your answer to $$2\ d.p.$$​

pH and weak acids

Weak acids and K_a

Weak acids don't dissociate fully in solution. This means the assumption that the acid concentration is the same as the hydrogen ion concentration can't be applied. An equilibrium constant is used instead. This is called the acid dissociation constant, $$K_a$$. 

Example

Derive the $$K_a$$ equation.

​

Write the equilibrium for a weak acid.

$$HA(aq)⇌H^+(aq)+A^-(aq)$$

Weak acids dissociate slightly in solution. This means assumption that the weak acid concentration is so much higher than the hydrogen ions concentration can be applied.

$$[HA]>>[H^+]$$

This suggests that the initial weak acid's concentration is the same as the equilibrium weak acid's concentration.

$$[HA]_{initial} \approx [HA]_{equilibrium}$$

Write the equilibrium constant.

$$K_a=\dfrac{[H^+][A^-]}{[HA]}$$

The assumption the acid dissociates more than water. This suggests that all the hydrogen ions in solution are from the acid instead of water.

$$[H^+(aq)]\approx[A^-(aq)]$$

Apply the assumption into the acid dissociation constant equation.

$$K_a=\dfrac{[H^+]^2}{[HA]}$$​​

The acid dissociation constant is $$K_a=\dfrac{[H^+]^2}{[HA]}$$.

Note: The units for $$K_a$$ are $$mol\>dm^{-3}$$.​

K_a and pH

The acid dissociation constant is very important to find the pH of a weak acid. The value of the dissociation constant is particular to the temperature. If the value of $$K_a$$ is known, the hydrogen ions concentration can be calculated. Using the calculated hydrogen ion concentration, the pH can be calculated.

Example

Calculate the pH of a $$0.03\>mol\>dm^{-3}$$ solution of benzoic acid, $$C_6H_5COOH$$. The $$K_a$$ for benzoic acid at this temperature is $$1.25\times10^{-5}\>mol\>dm^{-3}$$.

Write the equation for $$K_a$$.

$$K_a=\dfrac{[H^+]^2}{[HA]}$$​​

Substitute the known weak acid.

$$K_a=\dfrac{[H^+]^2}{[C_6H_5COOH]}$$

Rearrange for $$[H^+]^2$$.

$$[H^+]^2=[C_6H_5COOH]\times K_a$$​​

Insert the known $$[C_6H_5COOH] \>and\>K_a$$ values.

$$[H^+]^2=0.03\times 1.25\times10^{-5}=3.75\times10^{-7} {(\>mol\>dm^{-3})}^2$$

Square root $$[H^+]^2$$ to give $$[H^+]$$.

$$[H^+]=6.12\times10^{-4} \>mol\>dm^{-3}$$​​

​

Using the calculated $$[H^+]$$, the pH can be calculated using the pH equation. Write down the pH equation.

$$pH=-\log _{10}[H^+]$$

Insert known value of $$[H^+]$$.

$$pH=-\log _{10}[6.12\times10^{-4}]=3.21$$

The pH is $$\underline{3.21}$$.