Problem 36 - Chemistry and Chemical Reactivity

Question

Insulin is a protein important in the metabolism of sugar. Its molar mass can be determined by means of an osmotic pressure experiment. A \(50.0-\mathrm{mg}\) sample of insulin was dissolved in enough water to make 100. mL of solution. This solution was determined to have an osmotic pressure of \(21.8 \mathrm{mm}\) \(\mathrm{H}_{2} \mathrm{O} .\) What is the molar mass of insulin? (Hint: The density of mercury is \(13.6 \mathrm{g} / \mathrm{mL} .\) )

Step-by-Step Solution

Answer

The molar mass of insulin is approximately 5760 g/mol.

Step 1: Understanding Osmotic Pressure

Osmotic pressure (\( \pi \)) is calculated using the formula \( \pi = iMRT \), where \( i \) is the van 't Hoff factor, \( M \) is the molarity, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. For a non-electrolyte like insulin, \( i \) is 1.

Step 2: Calculate Osmotic Pressure in Standard Units

Convert the given osmotic pressure from mm Hg to atm: \[21.8 \text{ mm H}_2\text{O} \times \frac{1 \text{ mm Hg}}{13.6 \text{ mm H}_2\text{O}} \times \frac{1 \text{ atm}}{760 \text{ mm Hg}} = 0.00214 \text{ atm}.\]

Step 3: Set Up the Formula for Molarity

Using the osmotic pressure formula, rearrange for molarity,\( M = \frac{\pi}{RT} \). Use \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \), and assume room temperature,\( 298 \text{ K} \).

Step 4: Calculate Molarity

Substitute the given values into the molarity formula:\[M = \frac{0.00214 \text{ atm}}{0.0821 \text{ L atm K}^{-1} \times 298 \text{ K}} \approx 8.68 \times 10^{-5} \text{ mol/L}.\]

Step 5: Determine Moles of Insulin

Since molarity is moles per liter, calculate moles of insulin in the 0.100 L of solution: \[\text{moles} = 8.68 \times 10^{-5} \text{ mol/L} \times 0.100 \text{ L} = 8.68 \times 10^{-6} \text{ mol}.\]

Step 6: Calculate Molar Mass of Insulin

Using the formula for molar mass,\( \text{Molar mass} = \frac{\text{mass (g)}}{\text{moles}} \). Substitute the mass of insulin (0.0500 g) and the calculated moles:\[\text{Molar mass} = \frac{0.0500 \text{ g}}{8.68 \times 10^{-6} \text{ mol}} \approx 5760 \text{ g/mol}.\]

Key Concepts

Osmotic Pressure

Osmotic pressure is an essential concept when dealing with solutions involving water movement. It is the pressure required to stop the flow of water through a semipermeable membrane.

• Imagine two sides separated by a thin membrane, one with pure water and the other with a sugar solution. The water naturally moves from the pure side to the sugar side to balance the concentration.
• Osmotic pressure, denoted by \( \pi \), measures this tendency of water to move to the more concentrated side.

In mathematical terms, osmotic pressure for a solution can be described by the equation: \[ \pi = iMRT \]Here:

• \( i \) is the van 't Hoff factor, representing the number of particles the solute forms in solution. For insulin, this value is 1 since it's a non-electrolyte.
• \( M \) is the molarity or concentration of the solution.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature in Kelvin.

Understanding osmotic pressure is crucial in determining molar mass, as seen in the insulin example, where the measurement of pressure aids in calculating the insulin's concentration.

Ideal Gas Constant

The ideal gas constant, symbolized by \( R \), is an important element in chemical calculations. It bridges various gas laws to a common equation:\[ PV = nRT \]In this context, \( R \) helps connect pressure, volume, temperature, and moles in an ideal gas. It's a universal constant with several values depending on units used.

• For osmotic pressure problems like insulin, \( R \) is often expressed in \( 0.0821 \, \text{L atm K}^{-1} \text{ mol}^{-1} \).
• This form makes it compatible with calculations involving atmospheric pressure and temperature in Kelvin.

Choosing the correct value for \( R \) ensures correct calculation of molarity and ultimately helps determine properties like molar mass. This value is constant, allowing for consistency in calculations across different systems.

Molarity

Molarity is a term used to describe how concentrated a solution is. It refers to the number of moles of a solute, such as insulin, present in one liter of solution. It is crucial in calculating vital properties of a chemical compound.

• Molarity (\( M \)) is defined as the moles of solute divided by liters of solution.\[ M = \frac{\text{moles of solute}}{\text{liters of solution}} \]
• In the insulin example, the calculated molarity represents the concentration of insulin in the prepared solution.
• Using osmotic pressure, the molarity can be determined, which then allows us to find the number of moles in the solution.

Understanding molarity helps in many calculations related to chemical reactions and solution properties. It is a cornerstone concept for relating concentrations to other physical properties like molar mass.

Insulin

Insulin plays a significant role in regulating blood sugar levels. In scientific research, determining its molar mass is essential for understanding its biochemical properties and behavior. Determining the molar mass involves calculations using osmotic pressure experiments.

• Measuring the osmotic pressure of a known concentration of insulin solution provides data to calculate molarity.
• The molar mass is then determined by dividing the mass of insulin used by the number of moles calculated from molarity.
• The molar mass is crucial for various applications, from drug development to medical treatment plans.

Insulin's precise molar mass ensures that it can be studied and administered correctly, maintaining its vital role in health and medicine.