May I get help on this

Question

What are the molar concentrations of acetic acid (CH3COOH) and sodium acetate (CH3COONa) in an aqueous solution buffered at a pH of 4.06 that has a freezing point of −2.09 °C? Assume complete dissociation of sodium acetate and a density of 1.02 g/mL for the solution. The p𝐾a for CH3COOH is 4.75.

[CH3COOH]=M?

[CH3COONa]=M?

Answer 1

[CH₃COOH] = 0.767 M and [CH₃COONa] = 0.157 M.

Two pieces of information, two unknowns. The pH gives the ratio of the components, the freezing point gives the total amount of dissolved particles, and the density is what finally converts molality into molarity. Take them in that order.

1. Ratio, from Henderson-Hasselbalch.

4.06 = 4.75 + log([A^-]/[HA]), so log([A^-]/[HA]) = -0.69

[A^-]/[HA] = 10^-0.69 = 0.2042

The pH sits below the pK_a, so the acid form should dominate. It does, by about five to one, which is a useful early check that you have the ratio the right way up.

2. Total particles, from freezing point depression. With K_f = 1.86 °C/m for water:

m_{total particles} = ΔT_f/K_f = 2.09 / 1.86 = 1.1237 mol/kg

Count particles, not formula units. Sodium acetate dissociates completely, so each mole gives two, Na⁺ and CH₃COO^-. Acetic acid is weak and stays essentially intact at one. Writing a for acetic acid and b for sodium acetate, both in molal:

a + 2b = 1.1237, with b = 0.2042a

a(1 + 2 × 0.2042) = 1.1237, so a = 0.7979 m and b = 0.1629 m

3. Molality to molarity, using the density. This is the step the problem is really testing, and it is why the density was given at all. Take 1 kg of water as the basis:

mass of acetic acid = (0.7979)(60.05) = 47.91 g; mass of sodium acetate = (0.1629)(82.03) = 13.36 g

total solution mass = 1000 + 47.91 + 13.36 = 1061.3 g

volume = 1061.3 / 1.02 = 1040.5 mL = 1.0405 L

[CH₃COOH] = 0.7979 / 1.0405 = 0.767 M

[CH₃COONa] = 0.1629 / 1.0405 = 0.157 M

Note that the solution mass is the water plus the solutes. Dividing 1000 g by the density instead of 1061 g is the usual slip here, and it throws both answers off by about 6%.

Finally, check the ratio survived: 0.157 / 0.767 = 0.204, exactly the value step 1 required. That works because converting molality to molarity divides both species by the same volume, so the ratio cannot change. If yours does not reproduce, the arithmetic went wrong in step 3, not in step 1.