Question

A new enzyme is isolated and purified. Its molar mass is determined by measuring the osmotic pressure of \(0.850 \mathrm{~L}\) of an aqueous solution containing \(7.89 \mathrm{~g}\) of the enzyme. The osmotic pressure of the solution is \(1.14 \mathrm{~mm} \mathrm{Hg}\) at \(22^{\circ} \mathrm{C}\). What is the molar mass of the enzyme?

Short answer

Answer: The molar mass of the enzyme is approximately 136,034 g/mol.

Step-by-step solution

1. Convert temperature to Kelvin

To convert the temperature from Celsius to Kelvin, add 273.15 to the given temperature: \(T = 22^{\circ} \mathrm{C} + 273.15 = 295.15\ \mathrm{K}\).

2. Convert osmotic pressure to atm

Next, convert the osmotic pressure from mm Hg to atm using the conversion factor 1 atm = 760 mm Hg: \(p = \dfrac{1.14\ \mathrm{mm\ Hg}}{760\ \mathrm{mm\ Hg/atm}} = 0.0015\ \mathrm{atm}\).

3. Calculate the number of moles (n)

Rearrange the osmotic pressure formula to solve for the number of moles, n: \(n = \dfrac{pV}{RT}\). Now plug in the given values and the gas constant (R = 0.0821 L atm/(mol K)): \(n = \dfrac{0.0015\ \mathrm{atm} \times 0.850\ \mathrm{L}}{0.0821\ \mathrm{L\ atm/(mol\ K)} \times 295.15\ \mathrm{K}} = 0.000058\ \mathrm{mol}\).

4. Calculate the molar mass

Finally, calculate the molar mass using the mass of the enzyme and the number of moles: Molar mass = \(\dfrac{\mathrm{mass}}{\mathrm{moles}} = \dfrac{7.89\ \mathrm{g}}{0.000058\ \mathrm{mol}} = 136,034\ \mathrm{g/mol}\). The molar mass of the enzyme is approximately 136,034 g/mol.