Question

Catalase, a liver enzyme, dissolves in warer. A \(10.0-\mathrm{mL}\). solution containing \(0.166 \mathrm{~g}\) of catalase exhibits an osmotic pressure of \(1.2\) Torr at \(20 .{ }^{\circ} \mathrm{C}\). What is the molar mass of catalase? $.

Short answer

The molar mass of catalase is calculated by using the osmotic pressure and the mass of catalase in the solution.

Step-by-step solution

Convert osmotic pressure to atmospheres

Osmotic pressure is given in Torr, to use it in the formula for osmotic pressure, it needs to be converted to atmospheres (atm). Use the conversion: 1 atm = 760 Torr. So, the osmotic pressure in atm is given by dividing the osmotic pressure in Torr by 760.

Convert Celsius to Kelvin

To use the osmotic pressure formula, the temperature must be in Kelvin. Convert temperature from degrees Celsius to Kelvin by adding 273.15 to the Celsius temperature.

Use the van't Hoff equation

The van't Hoff equation relates osmotic pressure (\( \text{II} \) to the molar concentration (\( M \)), the universal gas constant (\( R \)), and the temperature (\( T \) in Kelvin). It is given by \( \text{II} = MRT \) where \( \text{II} \) is the osmotic pressure, \( M \) is the molarity of the solution, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Calculate Molarity

Rearrange the van't Hoff equation to solve for molarity (\( M \)): \( M = \frac{\text{II}}{RT} \). Use the converted osmotic pressure and temperature to calculate the molarity of the catalase solution.

Calculate moles of catalase

Once the molarity is calculated, find the number of moles of catalase using the volume of the solution. The number of moles (\( n \)) is equal to molarity (\( M \)) times the volume (\( V \)) in liters (L): \( n = MV \).

Calculate molar mass of catalase

Molar mass (\( M_m \)) is the mass of one mole of a substance. Calculate the molar mass of catalase using the mass of catalase (in grams) and the moles of catalase: \( M_m = \frac{\text{mass (g)}}{n} \).

Key concepts

Osmotic Pressure

Understanding osmotic pressure is crucial for determining the molar mass of a substance like catalase enzyme in a solution. Osmosis is the passage of a solvent, such as water, through a semipermeable membrane from a region of lower solute concentration to one of higher concentration. The osmotic pressure is the pressure required to prevent this flow of solvent. It's a colligative property, meaning it depends on the number of solute particles in the solution but not on their identity.

In the case of the catalase solution, the measured osmotic pressure provides insight into the concentration of the enzyme present. It's the driving force that allows us to calculate other important values down the line, such as the molarity of the catalase solution, eventually leading us to find its molar mass.

van't Hoff Equation

The van't Hoff equation plays a central role in connecting osmotic pressure with molarity. It states that osmotic pressure (\( \text{II} \)) for a dilute solution is directly proportional to the molarity (\( M \) of the solute, the gas constant (\( R \) and the absolute temperature (\( T \) in Kelvin. Mathematically, it's expressed as \( \text{II} = MRT \) .

For students, it's vital to understand how to manipulate this equation to find the molarity, which is a stepping stone in calculating molar mass. By rearranging the equation (\( M = \frac{\text{II}}{RT} \) we can find the molarity when osmotic pressure, temperature, and the gas constant are known. The equation emphasizes that as osmotic pressure increases, so does the concentration of the solute in the solution.

Molarity Calculation

Molarity is defined as the number of moles of solute per liter of solution. In this case, to calculate the molarity of the catalase solution, we rearrange the van't Hoff equation to solve for molarity:

\( M = \frac{\text{II}}{RT} \)

After plugging in the values for osmotic pressure (in atmospheres), temperature (in Kelvin), and the gas constant, we can calculate the molarity. This step is essential as molarity reflects the concentration of catalase in the solution, allowing further calculations to determine its molar mass.

Temperature Conversion

Temperature conversion from Celsius to Kelvin is a key step in working with the van't Hoff equation, which requires absolute temperature for accurate calculations. Since Kelvin is the SI unit of thermodynamic temperature and forms the basis for many scientific calculations, we add 273.15 to the Celsius temperature to convert it. For instance, to convert 20 degrees Celsius to Kelvin, we perform the following calculation:

\( T(K) = 20 ^{\text{°C}} + 273.15 = 293.15 \) K

Gas Constant

The gas constant (\( R \) is a fundamental constant in chemistry, appearing in many equations related to gases and thermodynamics, including the van't Hoff equation. It's the physical constant that relates energy scales to temperature scales and is typically given in units of liters atm per mol K (\( \text{L atm K}^{-1} \text{mol}^{-1} \) for context, its value is approximately 0.0821. In osmotic pressure calculations, using the correct value and units for the gas constant is crucial for obtaining the correct molarity of the solution.

Catalase Enzyme

Catalase is an enzyme found in the cells of many organisms, including humans. It's known for catalyzing the decomposition of hydrogen peroxide to water and oxygen, an important reaction that helps protect cells from oxidative damage. Understanding the properties of catalase, such as its molar mass, is beneficial for various scientific and medical fields. The exercise's aim to determine catalase's molar mass through osmotic pressure is an example of how biochemistry blends with physical chemistry to characterize biomolecules.