Question

An aqueous solution of 10.00 $\mathrm{g}$ of catalase, an enzyme found in the liver, has a volume of 1.00 $\mathrm{L}$ at ${27}^{\circ }\mathrm{C}$ . The solution's osmotic pressure at ${27}^{\circ }\mathrm{C}$ is found to be 0.745 torr. Calculate the molar mass of catalase.

Short answer

The molar mass of catalase is approximately $2.41\times {10}^5\mathrm{g}/\mathrm{mol}$.

Step-by-step solution

Convert the temperature to Kelvin and pressure to atm

First, we need to convert the given temperature from Celsius to Kelvin: $T(K)=T(°C)+273.15$ Next, we need to convert the osmotic pressure from torr to atm: 1 atm = 760 torr

Calculate the molar concentration of catalase

Equating the osmotic pressure formula, we get: Osmotic Pressure = (Molar Concentration) × (R) × (Temperature in K) Rearranging to find Molar Concentration: Molar Concentration = (Osmotic Pressure) / (R × Temperature in K) Now, substitute the given values and the converted values from Step 1 into the equation and solve for the molar concentration: R (ideal gas constant) = 0.0821 L atm / (K mol)

Find the number of moles of catalase

Using the calculated molar concentration and the volume of the solution, we can find the number of moles of catalase: Number of moles of catalase = (Molar Concentration) × (Volume of solution in L)

Calculate the molar mass of catalase

Finally, we can use the formula for molar mass to find the molar mass of catalase: Molar Mass = (mass of solute) / (number of moles of solute) Now, substitute the values obtained from Steps 2 and 3 into the formula and find the molar mass of catalase.

Key concepts

Molar Mass Calculation

When finding the molar mass of a compound, we're essentially discovering how much one mole of that compound weighs. Molar mass is expressed in grams per mole (g/mol). For the given exercise, calculating the molar mass involves a series of systematic steps based on other known values.
The process typically starts by determining the various quantities needed, such as mass and moles. In this particular problem, we start with a known mass of catalase, which is 10 grams. To find the molar mass, it's essential first to determine the number of moles of the substance.
We calculate the number of moles using the formula:

• Number of moles = Molar Concentration × Volume of solution (in liters)

Once the number of moles is calculated, it's straightforward to find the molar mass using the formula:

• Molar Mass = Mass of Solute / Number of Moles of Solute

By substituting the values found through this step-by-step process, we can derive the molar mass of catalase, which helps us understand its properties on a molecular level.

Ideal Gas Law

The Ideal Gas Law is fundamental in describing the state of an ideal gas. It's typically expressed as:

• $$PV=nRT$$

where:

• $P$ stands for pressure of the gas.
• $V$ is the volume the gas occupies.
• $n$ represents the number of moles of the gas.
• $R$ is the ideal gas constant (0.0821 L atm K⁻¹ mol⁻¹).
• $T$ is the temperature in Kelvin.

In the context of the osmotic pressure problem, the formula is adapted to the properties of solutions. Osmotic pressure ($\pi$) is analogous to the gaseous pressure and can be expressed as:

• $$\pi =iMRT$$

In this formula, $i$ is the van 't Hoff factor, which is integral for solutions with dissociative solutes, but equals 1 for non-dissociating solutes like proteins.
By rearranging the equation, you determine the molar concentration of the solute, which is crucial to finding other properties like molar mass. Understanding these concepts allows for practical application, such as working with enzyme solutions or other complex biological materials.

Enzyme Chemistry

Enzymes are proteins that accelerate chemical reactions and are fundamental to numerous biological processes. Catalase is a specific enzyme responsible for breaking down hydrogen peroxide into water and oxygen in the liver, preventing chemical harm to cells.
The study of enzyme chemistry includes understanding the structure, function, and mechanisms by which enzymes facilitate reactions. Enzymes operate under specific conditions of pH and temperature, acting as highly specialized catalysts.
In the given exercise, knowing the molar mass of an enzyme like catalase involves using biochemical and biophysical principles to quantify its molecular weight. This weight is essential for understanding how much enzyme is required for a specific reaction.
By leveraging basic chemistry laws like the Ideal Gas Law and understanding properties like osmotic pressure, scientists can make inferences about the behavior of enzymes in various environments. This knowledge is crucial in fields ranging from pharmaceuticals to biotechnology, where enzymes play pivotal roles in the development of new therapies and products.