Question

Lysozyme is an enzyme that breaks bacterial cell walls. A solution containing 0.150 gof 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 13,898.22 g/mol.

Step-by-step solution

Convert the temperature to Kelvin

Given the temperature is 25°C, we need to convert it to Kelvin by adding 273.15 to it. \(T(K) = 25 + 273.15 = 298.15\ K\)

Convert osmotic pressure to atm

The osmotic pressure is given in torr. We need to convert it to atm (the unit for the ideal gas constant R). 1 atm = 760 torr, therefore: \(\pi(atm) = \dfrac{0.953 \ torr}{760 \ torr/atm} = 0.001254 \ atm\)

Use the osmotic pressure equation

Rearrange the osmotic pressure equation to get the concentration: \((concentration) = \dfrac{\pi}{R \times T}\) Plug in the osmotic pressure, temperature, and the ideal gas constant (R = 0.0821 L atm/mol K): \((concentration) = \dfrac{0.001254 \ atm}{0.0821 \ L \ atm/mol \ K \times 298.15 \ K} = 5.14 \times 10^{-5} \ mol/L\)

Determine the moles of lysozyme

The volume of the solution is given in milliliters (210 mL). Convert it to liters: \(V = \dfrac{210 \ mL}{1000 \ mL/L} = 0.21\ L\) Now, find the moles of lysozyme using the concentration: \((moles) = (concentration) \times (volume) = 5.14 \times 10^{-5} \ mol/L \times 0.21 \ L = 1.08 \times 10^{-5} \ mol\)

Calculate the molar mass

Now that we have the moles of lysozyme, we can calculate its molar mass. We are given that the mass of lysozyme is 0.150 g. Use the relationship: \(Molar \ mass = \dfrac{mass}{moles}\) Plug in the values: \(Molar \ mass = \dfrac{0.150 \ g}{1.08 \times 10^{-5} \ mol} = 13898.22 \ g/mol\) So, the molar mass of lysozyme is approximately 13,898.22 g/mol.

Key concepts

Osmotic Pressure

Osmotic pressure is a key concept used to understand the properties of solutions, especially in chemistry and biology. It is the pressure required to prevent the inward flow of water through a semi-permeable membrane.

This is particularly important for biological systems where cell membranes act as semi-permeable barriers. Osmotic pressure can be calculated using the formula:
\[ \pi = iMRT \]
- \(\pi\) represents the osmotic pressure,- \(i\) is the van 't Hoff factor (which varies based on the solute),- \(M\) is the molarity of the solution,- \(R\) is the ideal gas constant (0.0821 L atm/mol K),- \(T\) is the temperature in Kelvin.

By understanding osmotic pressure, one can determine other physical properties of the solution, such as molar mass, if other values are known.

Temperature Conversion

Converting temperature from Celsius to Kelvin is a common task in scientific calculations. The Kelvin scale is an absolute temperature scale where 0 K is absolute zero, the theoretical point where molecular motion ceases.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature:
\[ T(K) = T(°C) + 273.15 \]
For example, if the temperature is 25°C, converting to Kelvin becomes:
\( T(K) = 25 + 273.15 = 298.15\ K \)
Conversion of temperature to Kelvin is crucial because many scientific formulas, including those involving the ideal gas constant and osmotic pressure, require Kelvin rather than Celsius.

Ideal Gas Constant

The ideal gas constant, often symbolized as \(R\), is a fundamental constant used in many equations including the Ideal Gas Law and calculations involving solutions and gases. It relates pressure, volume, temperature, and amount of substance in an ideal gas.

The most common value for \(R\) when dealing with pressure in atmospheres and volume in liters is:

• \(R = 0.0821 \ L \ atm/mol \ K\)

This value allows one to work seamlessly between different properties of gases and solutions.

Understanding and correctly using \(R\) is vital to engaging in calculations that include factors like osmotic pressure, where understanding the behavior of liquids and gases is essential.

Solution Volume Conversion

In chemistry, it’s often necessary to convert the volume of a solution from milliliters to liters for calculations involving molarity. This is because standard units in chemistry are in liters when using equations involving molarity or the ideal gas law.

The conversion is straightforward:
- Remember that 1 liter = 1000 milliliters.- To convert milliliters to liters, divide the volume in milliliters by 1000.
For example, if you have 210 mL of solution, convert it to liters as follows:
\[ V(L) = \dfrac{210 \ mL}{1000} = 0.21 \ L \]
Having the volume in liters is essential when applying it in calculations like determining molarity or using the equation for osmotic pressure. It ensures consistent units and accurate results.