Problem 81 - Chemistry: The Central Science

Question

Lysozyme is an enzyme that breaks bacterial cell walls. A solution containing \(0.150 \mathrm{~g}\) of this enzyme in \(210 \mathrm{~mL}\) of solution has an osmotic pressure of \(0.953\) torr at \(25^{\circ} \mathrm{C}\). What is the molar mass of lysozyme?

Step-by-Step Solution

Answer

The molar mass of lysozyme is approximately \(4670 \thinspace g \thinspace mol^{-1}\).

Step 1: Convert the given values to the appropriate units.

First, we need to convert the given values into the appropriate units. - Temperature: Convert Celsius to Kelvin - \(T (K) = 25^{\circ} C + 273.15 = 298.15 K \) - Volume: Convert mL to L - \(V = 210 mL = 0.210 L \)

Step 2: Rearrange the osmotic pressure formula to solve for n (moles of solute).

We can rearrange the osmotic pressure formula to solve for n by multiplying both sides by V and then dividing by R and T: \(n = π * V / (R * T)\)

Step 3: Calculate the number of moles of lysozyme (n).

Now, we can plug in the values we've converted and calculated into the formula: \(n = (0.953 \thinspace torr) * (0.210 \thinspace L) / (0.0821 \thinspace L \thinspace atm \thinspace mol^{-1} \thinspace K^{-1}) * (298.15 \thinspace K) \) Since we need the pressure to be in atm for the ideal gas constant R, we must convert torr to atm: 1 atm = 760 torr 0.953 torr * (1 atm / 760 torr) = 0.001253 atm Now, we can plug in the pressure in atm: \(n = (0.001253 \thinspace atm) * (0.210 \thinspace L) / (0.0821 \thinspace L \thinspace atm \thinspace mol^{-1} \thinspace K^{-1}) * (298.15 \thinspace K) \) \(n = 3.212 \times 10^{-5} \thinspace mol \)

Step 4: Calculate the molar mass of lysozyme.

We can now calculate the molar mass (M) of lysozyme by dividing the given mass (m) by the number of moles (n) we just calculated: \(M = \frac{m}{n}\) \(M = \frac{0.150 \thinspace g}{3.212 \times 10^{-5} \thinspace mol} = 4671.64 \thinspace g \thinspace mol^{-1}\)

Step 5: Round the answer to an appropriate number of significant figures.

Since the given values were all in three significant figures, we should round our answer to three significant figures as well: \(M ≈ 4670 \thinspace g \thinspace mol^{-1}\) The molar mass of lysozyme is approximately 4670 g/mol.