An aqueous solution containing 34.6 g of an unknown molecular nonelectrolyte compound in 142.0 g of water was found to have a freezing point of 1.2 degrees C. Calculate the molar mass.
A sample of steam with a mass of 0.553 g and at a temperature of 100 degrees C condenses into an insulated container holding 4.25 g of water at 4.0 degrees C. Assuming that no heat is lost to the surroundings, what will be the final temperature of the mixture?
To solve both of these problems, we can use the concept of colligative properties, which are properties of solutions that depend on the number of solute particles present, rather than their chemical identity.
1. Calculating Molar Mass:
In the first problem, we have an aqueous solution with a known mass of a nonelectrolyte compound and the freezing point depression. We can use the formula for freezing point depression (∆Tf) to calculate the molar mass (M) of the unknown compound.
The formula for freezing point depression is:
∆Tf = Kf * molality
where ∆Tf is the change in freezing point, Kf is the cryoscopic constant (a property of the solvent), and molality is the molal concentration of the solution.
First, we need to calculate the molality of the solution:
molality (m) = moles of solute / mass of solvent (in kg)
To calculate moles of solute, we can use the formula:
moles of solute = mass of solute / molar mass
Given:
Mass of solute = 34.6 g
Mass of solvent (water) = 142.0 g
∆Tf = 1.2 degrees C
Kf for water = 1.86 degrees C/m
First, convert the mass of solvent to kg:
Mass of solvent (water) = 142.0 g = 0.142 kg
Next, calculate the molality (m) using the formula:
molality (m) = moles of solute / mass of solvent
To find moles of solute, divide the mass of solute by its molar mass:
moles of solute = 34.6 g / molar mass
Substitute the known values into the formula for freezing point depression:
∆Tf = Kf * molality
Rearrange the formula to solve for the molar mass:
molar mass = 34.6 g / (molality * Kf)
2. Calculating Final Temperature:
In the second problem, we have a steam sample condensing into water. We can use the concept of heat transfer and the specific heat capacity (C) of water to calculate the final temperature.
The formula for heat transfer is:
Q = m * C * ∆T
where Q is the heat transfer, m is the mass of the substance, C is the specific heat capacity, and ∆T is the change in temperature.
First, we need to calculate the heat absorbed by the water when it is heated from 4.0 degrees C to the final temperature.
Using the formula for heat transfer for the water:
Q_water = m_water * C_water * ∆T_water
Given:
Mass of water (m_water) = 4.25 g
Specific heat capacity of water (C_water) = 4.18 J/g°C (approximately)
Change in temperature (∆T_water) = final temperature - initial temperature = T_final - 4.0
Next, we calculate the heat released by the steam when it condenses to water:
Q_steam = m_steam * C_steam * ∆T_steam
Given:
Mass of steam (m_steam) = 0.553 g
Specific heat capacity of steam (C_steam) = 2.03 J/g°C (approximately)
Change in temperature (∆T_steam) = initial temperature - final temperature = 100 - T_final
Since there is no heat lost to the surroundings, the heat absorbed by the water is equal to the heat released by the steam:
Q_water = Q_steam
Substitute the known values into the heat transfer equation for water and steam:
m_water * C_water * ∆T_water = m_steam * C_steam * ∆T_steam
Simplify the equation and solve for the final temperature:
m_water * C_water * ∆T_water = m_steam * C_steam * ∆T_steam
(4.25 g) * (4.18 J/g°C) * (T_final - 4.0) = (0.553 g) * (2.03 J/g°C) * (100 - T_final)
Rearrange the equation and solve for T_final.