To calculate the pH of a buffer solution made from \( KH_2PO_4 \) (potassium dihydrogen phosphate) and \( Na_2HPO_4 \) (disodium hydrogen phosphate), we can use the Henderson-Hasselbalch equation.
Step 1: Calculate the number of moles of each component
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Moles of \( KH_2PO_4 \):
- Molar mass of \( KH_2PO_4 \): \[ K = 39.10 , \text{g/mol}, , H = 1.01 , \text{g/mol}, , P = 30.97 , \text{g/mol}, , O = 16.00 , \text{g/mol} \] \[ \text{Molar mass} = 39.10 + 2(1.01) + 30.97 + 4(16.00) = 136.09 , \text{g/mol} \]
- Moles of \( KH_2PO_4 \): \[ \text{Moles} = \frac{18.0 , \text{g}}{136.09 , \text{g/mol}} \approx 0.132 , \text{mol} \]
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Moles of \( Na_2HPO_4 \):
- Molar mass of \( Na_2HPO_4 \): \[ Na = 22.99 , \text{g/mol}, , H = 1.01 , \text{g/mol}, , P = 30.97 , \text{g/mol}, , O = 16.00 , \text{g/mol} \] \[ \text{Molar mass} = 2(22.99) + 1.01 + 30.97 + 4(16.00) = 141.96 , \text{g/mol} \]
- Moles of \( Na_2HPO_4 \): \[ \text{Moles} = \frac{34.0 , \text{g}}{141.96 , \text{g/mol}} \approx 0.239 , \text{mol} \]
Step 2: Find the concentrations in a 1.00 L solution
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Concentration of \( KH_2PO_4 \): \[ [KH_2PO_4] = \frac{0.132 , \text{mol}}{1.00 , \text{L}} = 0.132 , \text{M} \]
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Concentration of \( Na_2HPO_4 \): \[ [Na_2HPO_4] = \frac{0.239 , \text{mol}}{1.00 , \text{L}} = 0.239 , \text{M} \]
Step 3: Use the Henderson-Hasselbalch equation
The Henderson-Hasselbalch equation is given by: \[ pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right) \] Where:
- \( [A^-] \) is the concentration of the base form \( Na_2HPO_4 \).
- \( [HA] \) is the concentration of the acid form \( KH_2PO_4 \).
- \( pK_a \) of \( H_2PO_4^- \) (the conjugate acid of \( HPO_4^{2-} \)) is approximately 7.2 (you may use a more precise value based on specific literature).
Step 4: Substitute values into the equation
\[ pH = 7.2 + \log\left(\frac{0.239}{0.132}\right) \] Calculating the ratio: \[ \frac{0.239}{0.132} \approx 1.808 \] Now calculating the logarithm: \[ \log(1.808) \approx 0.257 \] Now, substituting back into the equation: \[ pH = 7.2 + 0.257 \approx 7.457 \]
Final Result
The pH of the buffer solution is approximately \( 7.46 \).
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