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# Calculating pH of Weak Acid and Base Solutions

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Calculating pH of Weak Acid and Base Solutions

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### Calculating pH of Weak Acid and Base Solutions

The  $K_a$ and  $K_b$ values have been determined for a great many acids and bases, as shown in Tables 21.5 and 21.6. These can be used to calculate the pH of any solution of a weak acid or base whose ionization constant is known.

#### Sample Problem: Calculating the pH of a Weak Acid

Calculate the pH of a 2.00 M solution of nitrous acid (HNO 2 ). The  $K_a$ for nitrous acid is 4.5 × 10 -4 .

Step 1: List the known values and plan the problem.

Known

• initial [HNO 2 ] = 2.00 M
• $K_a=4.5 \times 10^{-4}$

Unknown

• pH = ?

First, an ICE table is set up with the variable  $x$ used to signify the change in concentration of the substance due to ionization of the acid. Then the  $K_a$ expression is used to solve for  $x$ and calculate the pH.

Step 2: Solve.

 Concentrations [HNO 2 ] [H + ] [NO 2 − ] Initial 2.00 0 0 Change $-x$ $+x$ $+x$ Equilibrium $2.00-x$ $x$ $x$

The  $K_a$ expression and value is used to set up an equation to solve for $x$ .

$K_a=4.5 \times 10^{-4}=\frac{(x)(x)}{2.00 - x}=\frac{x^2}{2.00 - x}$

The quadratic equation is required to solve this equation for $x$ . However, a simplification can be made because of the fact that the extent of ionization of weak acids is small. The value of  $x$ will be significantly less than 2.00, so the " $-x$ ” in the denominator can be dropped.

$4.5 \times 10^{-4}&=\frac{x^2}{2.00 - x} \approx \frac{x^2}{2.00} \\x&=\sqrt{4.5 \times 10^{-4}(2.00)}=2.9 \times 10^{-2} \ \text{M}= \left [ H^+ \right ]$

Since the variable  $x$ represents the hydrogen-ion concentration, the pH of the solution can now be calculated.

$pH=- \log[\text{H}^+]=- \log[2.9 \times 10^{-2}]=1.54$

The pH of a 2.00 M solution of a strong acid would be equal to $- \log(2.00) = -0.30$ . The higher pH of the 2.00 M nitrous acid is consistent with it being a weak acid and therefore not as acidic as a strong acid would be.

The procedure for calculating the pH of a solution of a weak base is similar to that of the weak acid in the sample problem. However, the variable  $x$ will represent the concentration of the hydroxide ion. The pH is found by taking the negative logarithm to get the pOH, followed by subtracting from 14 to get the pH.

#### Summary

• The procedure for calculating the pH of a weak acid or base is illustrated.

#### Practice

Perform the calculations at the site below:

#### Review

1. What does  $x$ stand for in the equation?
2. What simplifying assumption is made?
3. What would  $x$ stand for if we were calculating pOH?