Question

A chemist wishes to determine the molecular weight and molecular formula of fructose (a sugar). He places \(.946 \mathrm{~g}\) of it in \(150 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{O}\) (water) and finds that the freezing point of water is depressed to \(-0.0651^{\circ} \mathrm{C} .\) Determine the molecular weight and formula of fructose, assuming that the simplest formula of fructose is \(\left(\mathrm{CH}_{2}\right) \mathrm{O}\).

Short answer

The molecular weight of fructose is approximately \(180.19~g~mol^{-1}\) and the molecular formula is \(C_6H_{12}O_6\).

Step-by-step solution

Find the Molality of the Fructose Solution

First, we need to find the molality of the fructose solution using the freezing point depression formula: \(ΔT_f = K_f * m\) Where: \(ΔT_f\) = freezing point depression = \(0.0 - (-0.0651) = 0.0651^{\circ}C\) \(K_f\) = cryoscopic constant of water = \(1.86^{\circ}C~kg~mol^{-1}\) \(m\) = molality of the fructose solution (in moles of solute per kg of solvent) Now, we will rearrange the formula to find the molality: \(m = \frac{ΔT_f}{K_f}\)

Calculate the Molality

Now, let's calculate the molality of the fructose solution: \(m = \frac{0.0651^{\circ}C}{1.86^{\circ}C~kg~mol^{-1}} = 0.0350~mol~kg^{-1}\)

Determine the Moles of Fructose

Using the molality formula, we can find the moles of fructose in the solution: \(m = \frac{moles~of~fructose}{mass~of~water~in~kg}\) Now, rearrange the formula to find the moles of fructose: \(moles~of~fructose = m * mass~of~water~in~kg\) \(moles~of~fructose = 0.0350~mol~kg^{-1} * 0.150~kg = 0.00525~mol\)

Calculate the Molecular Weight of Fructose

Now, we will use the moles of fructose and the mass of fructose to calculate the molecular weight: \(Molecular~weight~of~fructose = \frac{mass~of~fructose}{moles~of~fructose}\) \(Molecular~weight~of~fructose = \frac{0.946~g}{0.00525~mol} = 180.19~g~mol^{-1}\)

Identify the Molecular Formula of Fructose

We know that the simplest formula of fructose is (CH2O), with a molar mass of approximately \(12.01~u + 2 * 1.01~u + 16.00~u = 30.03~u\). Now, we will use the molecular weight (180.19 g/mol) to identify the molecular formula: \(Molecular~formula~multiplier = \frac{molecular~weight}{simplest~formula~weight}\) \(Molecular~formula~multiplier = \frac{180.19~g~mol^{-1}}{30.03~g~mol^{-1}} ≈ 6\) Since the multiplier is approximately 6, the molecular formula of fructose is: \(C_6H_{12}O_6\)

Key concepts

Freezing point depression

When a solute is added to a solvent, the freezing point of the solvent is lowered. This phenomenon is known as freezing point depression. It occurs because the presence of solute particles interferes with the formation of a solid structure, requiring a lower temperature for solidification. This is a colligative property, meaning it depends on the number of solute particles, not their identity.

In our exercise, fructose is the solute, and water is the solvent. The observed freezing point depression is determined by the difference between the pure solvent's freezing point and the solution's freezing point, calculated as:

• Freezing Point Depression (\( \Delta T_f \)) = Pure solvent's freezing point - Solution's freezing point
• For water: \( \Delta T_f = 0.0^{\circ}C - (-0.0651^{\circ}C) = 0.0651^{\circ}C \)

Cryoscopic constant

The cryoscopic constant (\( K_f \)) is a property specific to each solvent, representing the freezing point depression caused by a one-molal concentration of a non-volatile solute. It provides a direct way to calculate the molality when the freezing point depression is known.

For water, \( K_f \) is known to be \( 1.86^{\circ}C \cdot kg \cdot mol^{-1} \). This means that for each mole of solute per kilogram of water, the freezing point will lower by \( 1.86^{\circ}C \). Using \( K_f \) simplifies determining molality, which is a crucial step in figuring out other properties of the solution.

• In our exercise: \( \Delta T_f \) (from above) and \( K_f \) are used in the formula \( m = \frac{\Delta T_f}{K_f} \) to determine molality.

Molality calculation

Molality (\( m \)) is a concentration term describing the moles of solute per kilogram of solvent. It is different from molarity, which is based on the solution's volume. Molality is particularly useful for colligative properties since it doesn't change with temperature.

In this scenario, we calculate the molality of the fructose solution using the freezing point depression formula:

• Rearrange: \( m = \frac{\Delta T_f}{K_f} \)
• Calculate: \( m = \frac{0.0651^{\circ}C}{1.86^{\circ}C \cdot kg \cdot mol^{-1}} = 0.0350 \ mol \cdot kg^{-1} \)

This result shows how concentrated the sugar solution is, leading to further calculations of moles and molecular weight.

Molecular formula identification

Identifying the molecular formula of a compound requires the molecular weight and the simplest formula (empirical formula). The simplest formula gives the basic ratio of atoms. The molecular formula shows the actual number of atoms.

For fructose, the simplest formula is given as (CH\(_2\)O). The simplest formula's weight is about \( 30.03 \ g \cdot mol^{-1} \). To find the molecular formula, we divide the experimentally calculated molecular weight by the simplest formula's weight and find a multiplier.

• Multiplier = \( \frac{180.19 \ g \cdot mol^{-1}}{30.03 \ g \cdot mol^{-1}} \approx 6 \)

Consequently, multiply the empirical formula by 6:

• Molecular formula = \( C_6H_{12}O_6 \)

Fructose is therefore confirmed as a common sugar molecule.