Preparing a Phosphate Buffer for a Temperature-Sensitive Enzyme Assay
A research assistant must prepare 500 mL of 0.10 M sodium phosphate buffer at pH 7.40 for an enzyme assay. The laboratory has solid NaH2PO4 (MW 119.98) and Na2HPO4 (MW 141.96); the second dissociation pKa is 7.20 at 25 °C. The assay is run at 37 °C, and the protocol warns that ionic strength above 0.15 M inhibits the enzyme. A colleague proposes making a 1.0 M stock at pH 7.40 and diluting it tenfold on the day of use. The assistant also plans to add 1.0 mL of 0.10 M HCl to 500 mL of the buffer to test whether the pH is stable. The laboratory's pH meter is calibrated at 25 °C, and the assistant has only 30 min before the assay. The supervisor asks for the exact masses, a judgement about the dilution plan, and a prediction of the acid challenge, with reasoning rather than a recipe.
1. Calculate the masses of NaH2PO4 and Na2HPO4 needed. Show the Henderson-Hasselbalch setup and the final masses.
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Use Henderson-Hasselbalch: pH = pKa + log([HPO4^2-]/[H2PO4^-]). 7.40 = 7.20 + log R, so log R = 0.20 and R = 1.585. Total phosphate = 0.10 mol/L x 0.500 L = 0.050 mol. Let acid = x and base = 1.585x; then 2.585x = 0.050, x = 0.01934 mol acid and 0.03066 mol base. Mass NaH2PO4 = 0.01934 x 119.98 = 2.32 g. Mass Na2HPO4 = 0.03066 x 141.96 = 4.35 g. Dissolve separately, combine, and make to 500 mL. Check the label: if the dibasic salt is the heptahydrate (MW about 268.07), the mass must be adjusted; using the anhydrous MW would give the wrong concentration. Adjust final pH with small amounts of stock acid or base, not with strong HCl/NaOH if ionic strength must stay low.
2. Evaluate the colleague's plan to make a 1.0 M stock at pH 7.40 and dilute it tenfold. Will the pH be exactly 7.40? What about buffer capacity and ionic strength?
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In an ideal solution, a tenfold dilution preserves the base/acid ratio and therefore the pH; the measured pH may shift slightly because activity coefficients change with ionic strength. The buffer capacity, however, falls roughly tenfold. A 1.0 M stock is a reasonable storage form, but the diluted 0.10 M buffer will resist added acid or base much less. Ionic strength also matters: 0.10 M phosphate is near 0.11 M, below the 0.15 M limit, but adding NaCl or pH-adjusting with NaOH can push it higher. The plan is acceptable only if dilution is volumetric and the final pH is checked. Do not assume the pH set at 25 °C is 7.40 at 37 °C; phosphate pKa2 falls with temperature, so the actual pH will be slightly lower. Calibrate and measure at the assay temperature, or equilibrate the buffer at 37 °C before adjusting.
3. Predict the pH after adding 1.0 mL of 0.10 M HCl to 500 mL of the buffer, and compare with adding the same HCl to 500 mL of water. What does this show about buffer capacity?
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Moles HCl = 0.0010 L x 0.10 M = 1.0 x 10^-4 mol. This converts an equal amount of HPO4^2- to H2PO4^-: new base = 0.03066 - 0.00010 = 0.03056 mol; new acid = 0.01934 + 0.00010 = 0.01944 mol. Ratio = 1.572; pH = 7.20 + log(1.572) = 7.396. The change is only about 0.004 pH unit. In 500 mL of pure water, the same HCl gives about 2.0 x 10^-4 M H+ (neglecting water autoionization), pH about 3.70. The buffer therefore works, but this small acid challenge uses only a fraction of its capacity; add more acid and the pH will eventually fall steeply near the buffer's capacity limit. Also note the calculation uses the 25 °C pKa; at 37 °C the starting pH is slightly different.