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?

Short answer

The molar mass of lysozyme is approximately \(14053 ~g/mol\).

Step-by-step solution

Write down the given information

- Mass of lysozyme: \(0.150~g\) - Volume of solution: \(210~mL\) - Osmotic pressure: \(0.953~torr\) - Temperature: \(25^{\circ}C\)

Convert the given units to their appropriate units

We need to convert the given units to their appropriate units to use the osmotic pressure formula. We will convert the volume of the solution to liters and the osmotic pressure to atmospheres. - Volume of solution: \(210~mL = 0.210~L\) - Osmotic pressure: \(0.953~torr × \frac{1 ~atm}{760 ~torr} = 0.001253~atm\) We also need to convert the temperature to Kelvin: - Temperature: \(25^{\circ}C + 273.15 = 298.15~K\)

Use the osmotic pressure formula to calculate the molarity

The formula for osmotic pressure is given by: \(Π = MRT\), where \(Π\) is the osmotic pressure, \(M\) is the molarity, \(R\) is the gas constant (\(0.0821 \frac{L\times atm}{mol\times K}\)), and \(T\) is the temperature in Kelvin. Rearrange the formula for molarity: \(M = \frac{Π}{RT}\) Now, substitute the values: \(M = \frac{0.001253~atm}{(0.0821~\frac{L \times atm}{mol \times K})(298.15~K)}\)

Calculate the molarity

Calculate the molarity: \(M = 5.086 \times 10^{-5} ~mol/L\)

Calculate the molar mass

We have the mass of lysozyme and the molarity. We can now calculate the molar mass using the formula: \(Molar~mass=\frac{mass}{moles}\). We know the moles can be calculated as \(moles = M \times V\), where \(V\) is the volume of the solution. Substitute the values: \(moles = (5.086 \times 10^{-5} ~mol/L)(0.210 ~L) = 1.068 × 10^{-5}~mol\) Next, calculate the molar mass using the mass of lysozyme and the calculated moles: \(Molar~mass = \frac{0.150~g}{1.068 × 10^{-5} ~mol} = 14052.91 ~g/mol\) The molar mass of lysozyme is approximately \(14053 ~g/mol\).