Question

A biochemical engineer isolates a bacterial gene fragment and dissolves a 10.0 -mg sample in enough water to make \(30.0 \mathrm{~mL}\) of solution. The osmotic pressure of the solution is 0.340 torr at \(25^{\circ} \mathrm{C}\). (a) What is the molar mass of the gene fragment? (b) If the solution density is \(0.997 \mathrm{~g} / \mathrm{mL}\), how large is the freezing point depression for this solution \(\left(K_{\mathrm{f}}\right.\) of water \(\left.=1.86^{\circ} \mathrm{C} / \mathrm{m}\right) ?\)

Short answer

The molar mass of the gene fragment is calculated from osmotic pressure and solution volume. The freezing point depression is calculated using solution density, molality, and given freezing point depression constant.

Step-by-step solution

- Convert Given Units

Convert the osmotic pressure from torr to atm. Use the conversion: 1 atm = 760 torr Thus, 0.340 torr = \(\frac{0.340}{760} \text{ atm} \)

- Use the Osmotic Pressure Formula

Use the formula \( \Pi = MRT \) where \(\Pi\) is the osmotic pressure, \(M\) is the molarity, \(R\) is the gas constant (0.0821 L atm / K mol), and \(T\) is the temperature in Kelvin. First, convert the temperature from Celsius to Kelvin: \( T = 25 + 273.15 = 298.15 \text{ K} \)Then, solve for molarity \(M\): \( 0.340 \text{ torr} = \frac{0.340}{760} \text{ atm} = M \cdot 0.0821 \text{ L atm / K mol} \cdot 298.15 \text{ K} \)

- Calculate Molarity

Rearrange to solve for \(M\): \( M = \frac{\Pi}{RT} \)Substitute the values: \( M = \frac{\frac{0.340}{760} \text{ atm}}{0.0821 \text{ L atm / K mol} \times 298.15 \text{ K}} \)

- Calculate Moles of Solute

Molarity is moles of solute over liters of solution. Use the volume of the solution to find the moles of solute: \( M = \frac{\text{moles of solute}}{0.030 \text{ L}} \)Solve for moles of solute: \( \text{moles of solute} = M \times 0.030 \text{ L} \)

- Find Molar Mass of Gene Fragment

Molar mass is the mass of the sample divided by the moles of solute. Given 10.0 mg sample (convert to grams): \( 10.0 \text{ mg} = 0.010 \text{ g} \)Molar mass = \( \frac{0.010 \text{ g}}{\text{moles of solute}} \)

- Use the Density for Freezing Point Depression

Given the density of the solution, find the mass of the solvent (water). Mass of solution = \( 30.0 \text{ mL} \times 0.997 \text{ g/mL} = 29.91 \text{ g} \)Subtract the mass of the gene fragment: Mass of solvent = 29.91 g - 0.010 g = 29.90 g

- Calculate Molality

Molality \(m\) is moles of solute per kilogram of solvent. Convert solvent mass to kilograms: \( 29.90 \text{ g} = 0.02990 \text{ kg} \)Molality \(m = \frac{\text{moles of solute}}{0.02990 \text{ kg}} \)

- Calculate Freezing Point Depression

Use the freezing point depression formula: \( \Delta T_f = K_f \cdot m \)Given \(K_f = 1.86^{\circ} \text{C/m} \), substitute the values to find the decrease in freezing point.

Key concepts

Osmotic Pressure

Osmotic pressure is the pressure required to stop the flow of solvent into a solution through a semipermeable membrane. It is directly related to the concentration of solute particles in the solution. The formula to calculate osmotic pressure is:
\( \Pi = MRT \), where:

• \( \Pi \) is the osmotic pressure.
• \( M \) is molarity (moles of solute per liter of solution).
• \( R \) is the gas constant (0.0821 L atm/K mol).
• \( T \) is the temperature in Kelvin.

To solve a problem involving osmotic pressure, you need to convert all units properly. For example, if the pressure is given in torr, convert it into atm first because the gas constant \( R \) is usually in L atm / K mol.
This conversion can be done using the equation:
\( 0.340 \text{ torr} = \frac{0.340}{760} \text{ atm} \).
Don't forget to convert the temperature from Celsius to Kelvin by adding 273.15. Understanding and accurately applying these conversions is crucial for correct calculations.

Freezing Point Depression

Freezing point depression is a colligative property. It describes the lowering of the freezing point of a solvent when a solute is dissolved in it. The change in freezing point \( \Delta T_f \) can be calculated using the formula:
\( \Delta T_f = K_f \cdot m \), where:

• \( K_f \) is the freezing point depression constant of the solvent.
• \( m \) is the molality of the solution.

Molality (m) is defined as the moles of solute per kilogram of solvent. To determine the actual freezing point depression, you need to know both the amount of solute and the amount of solvent.
For example, in the provided problem, the mass of the gene fragment and the solution's volume are given, thus, you can calculate the molality by first finding moles of solute and then dividing by the mass of the solvent after converting it to kilograms.
These key steps ensure accurate determination of the freezing point depression.

Molarity

Molarity (M) is a way to express the concentration of a solution. It is defined as the number of moles of solute per liter of solution. The formula is:
\( M = \frac{\text{moles of solute}}{\text{liters of solution}} \).
To find the molarity, ensure you correctly convert all given units. For example, converting milliliters to liters and grams to moles (using the molar mass of the solute).
Start with the basics:
1. Convert the volume of the solution to liters if it's in milliliters:

• \(30 \text{ mL} = 0.030 \text{ L} \).

2. Find the number of moles of the solute. In this example, you would use the formula for osmotic pressure to rearrange and solve for molarity, then use the volume to find moles.
Mastery of molarity concepts is essential for chemical calculations involving solutions, reactions, and properties like osmotic pressure or freezing point depression.