Phosphoric Acid Buffer Ratio Calculation

To determine the required ratio of $Na_2HPO_4$ to $NaH_2PO_4$ for a buffer solution at a specific pH, we use the Henderson-Hasselbalch equation.

Henderson-Hasselbalch Equation Application

The relevant form of the Henderson-Hasselbalch equation for a buffer system is:

$ pH = pK_a + log \frac{[Conjugate \ Base]}{[Conjugate \ Acid]} $

In this case:

• The target buffer pH is $7.0$.
• The relevant pKa is the second dissociation constant of phosphoric acid, $pK_{a2} = 6.8$.
• The conjugate base is $HPO_4^{2-}$ (provided by $Na_2HPO_4$).
• The conjugate acid is $H_2PO_4^-$ (provided by $NaH_2PO_4$).

Calculating the Ratio

Substitute the known values into the equation:

$ 7.0 = 6.8 + log \frac{[Na_2HPO_4]}{[NaH_2PO_4]} $

Rearrange the equation to solve for the logarithm of the ratio:

$ log \frac{[Na_2HPO_4]}{[NaH_2PO_4]} = 7.0 - 6.8 $

$ log \frac{[Na_2HPO_4]}{[NaH_2PO_4]} = 0.2 $

To find the ratio, take the antilogarithm (10 raised to the power of 0.2):

$ \frac{[Na_2HPO_4]}{[NaH_2PO_4]} = 10^{0.2} $

Calculate the value:

$ 10^{0.2} \approx 1.58489 $

Rounding the result to two decimal places gives:

$ \frac{[Na_2HPO_4]}{[NaH_2PO_4]} \approx 1.58 $

Therefore, the ratio of $Na_2HPO_4$ to $NaH_2PO_4$ required is approximately 1.58.