A biochemical engineer isolates a bacterial gene fragment and dissolves a 11.8 mg sample of the material in enough water to make 36.1 mL of solution. The osmotic pressure of the solution is 0.340 torr at 25°C.

(a) What is the molar mass of the gene fragment?(in g/mol)
(b) If the solution density is 0.997 g/mL, how large is the freezing point depression for this solution (Kf of water=1.86°C/m)? (in degrees C)

pi = MRT

Use pi in atm; i.e., 340/760 = ?atm.
Solve for M = molarity.
Then M = mols/L soln. You know L and you know M, solve for mols.
Then mol = grams/molar mass. You know grams and mols, solve for molar mass.

b.
delta T = Kf*m
You know M, you want m.
Use density to convert volume to grams of solution, and recognize that grams soln = g solute + grams solvent. Subtract to find grams solvent, convert to kg then to m and calculate delta T.

To find the molar mass of the gene fragment, we can use the equation for osmotic pressure:

Π = MRT

Where:
Π = osmotic pressure
M = molarity
R = gas constant (0.0821 atm·L/mol·K or 8.314 J/mol·K)
T = temperature in Kelvin

(a) First, let's calculate the molarity (M) of the solution:

Molarity (M) = moles of solute / volume of solution in liters

Given:
Mass of gene fragment (solute) = 11.8 mg
Volume of solution = 36.1 mL = 0.0361 L

Molarity (M) = (11.8 mg / molar mass) / 0.0361 L

Since we need the molar mass, we can rearrange the equation as follows:

molar mass = (11.8 mg / (Molarity × volume of solution))

Substituting the given values, we get:

molar mass = (11.8 mg / (0.340 torr × 0.0821 atm·L/mol·K × (25 + 273) K × 0.0361 L))

Now we need to convert the mass from mg to grams:

molar mass = (0.0118 g / (0.340 torr × 0.0821 atm·L/mol·K × (25 + 273) K × 0.0361 L))

Calculating this expression will give the molar mass of the gene fragment in g/mol.

(b) To find the freezing point depression, we can use the equation:

ΔTf = Kf × m

Where:
ΔTf = freezing point depression
Kf = cryoscopic constant for water (1.86°C/m)
m = molality of the solution

Molality (m) can be calculated using the formula:

Molality (m) = moles of solute / mass of solvent in kg

Given:
Mass of solvent (water) = volume of solution × density of solution
Volume of solution = 36.1 mL = 0.0361 L
Density of solution = 0.997 g/mL

Mass of solvent (water) = 0.0361 L × 0.997 g/mL

Molality (m) = moles of solute / (0.0361 L × 0.997 g/mL) × (1 kg/1000 g)

Again, since we know the mass of the gene fragment, we can rearrange the equation as:

ΔTf = Kf × (moles of solute / (0.0361 L × 0.997 g/mL) × (1 kg/1000 g))

Calculating this expression will give the freezing point depression in degrees Celsius.

To solve these questions, we need to use the concepts of osmotic pressure and freezing point depression.

(a) To determine the molar mass of the gene fragment, we first need to convert the given data into the number of moles of the solute.

Step 1: Convert the mass of the solute (11.8 mg) into grams:
11.8 mg = 0.0118 g

Step 2: Calculate the number of moles using the formula:
moles = mass / molar mass

In this case, we have:
0.0118 g / molar mass = moles

Now, let's calculate the number of moles.

Step 3: Convert the volume of the solution from milliliters to liters:
36.1 mL = 0.0361 L

Step 4: Use the formula for osmotic pressure to relate moles of solute to the osmotic pressure:
osmotic pressure = (moles / volume) * R * temperature

Where:
R = ideal gas constant (0.0821 L·atm/(mol·K))
temperature = 25°C = 298 K

Rearranging the formula, we get:
moles = (osmotic pressure * volume) / (R * temperature)

Plugging in the values, we have:
moles = (0.340 torr * 0.0361 L) / (0.0821 L·atm/(mol·K) * 298 K)

Simplifying the expression, we get:
moles ≈ 0.00524 mol

Step 5: Calculate the molar mass of the gene fragment using the formula:
molar mass = mass / moles

Plugging in the values, we have:
molar mass ≈ 0.0118 g / 0.00524 mol

Simplifying the expression, we get:
molar mass ≈ 2.25 g/mol

Therefore, the molar mass of the gene fragment is approximately 2.25 g/mol.

(b) To determine the freezing point depression, we need to use the relationship between molality and the freezing point depression.

Step 1: Calculate the molality of the solution using the formula:
molality = moles of solute / mass of solvent (in kg)

In this case, the mass of solvent can be calculated using the given density:
mass of solvent = volume of solution * density

Plugging in the values, we have:
mass of solvent = 0.0361 L * 0.997 g/mL

Simplifying the expression, we get:
mass of solvent ≈ 0.0360 kg

Now, let's calculate the molality:
molality = 0.00524 mol / 0.0360 kg

Simplifying the expression, we get:
molality ≈ 0.145 mol/kg

Step 2: Use the formula for freezing point depression to calculate the change in freezing point:
ΔT = Kf * molality

Plugging in the values, we have:
ΔT = 1.86 °C/m * 0.145 mol/kg

Simplifying the expression, we get:
ΔT ≈ 0.27 °C

Therefore, the freezing point depression for this solution is approximately 0.27 °C.