Question

A sugar solution was prepared by dissolving \(9.0 \mathrm{~g}\) of sugar in \(500 \mathrm{~g}\) of water. At \(27^{\circ} \mathrm{C}\), the osmotic pressure was measured as \(2.46\) atm. Determine the molecular weight of the sugar.

Short answer

The molecular weight of the sugar is approximately \(180.49 \: g/mol\).

Step-by-step solution

Identify the osmotic pressure formula

The osmotic pressure formula is given by: \[P = cRT\] where: - \(P\) = osmotic pressure (atm) - \(c\) = molality of the solution (moles of solute/kg of solvent) - \(R\) = ideal gas constant (0.0821 L atm/mol K) - \(T\) = temperature (K)

Convert temperature to Kelvin

The given temperature is \(27^{\circ} \mathrm{C}\). To convert it to Kelvin, we add 273.15: \[T = 27 + 273.15 = 300.15 \: K\]

Find the molality

First, let's find the molality (c) of the solution. Molality is the moles of solute divided by the mass of solvent in kilograms: \[c = \frac{n_\text{sugar}}{m_\text{water}}\] where: - \(n_\text{sugar}\) = moles of sugar - \(m_\text{water}\) = mass of water in kg The mass of water is given as 500 g, so convert it to kg: \[m_\text{water} = 500 \: g = 0.5 \: kg\] We don't have the moles of sugar, so we'll rewrite the molality formula in terms of mass and molecular weight: \[c = \frac{m_\text{sugar}}{M_\text{sugar} \cdot m_\text{water}}\] where: - \(m_\text{sugar}\) = mass of sugar (9 g) - \(M_\text{sugar}\) = molecular weight of sugar

Rearrange the osmotic pressure formula

Now that we have the formula for the molality, we'll substitute it into the osmotic pressure formula and solve for the molecular weight of sugar: \[P = \frac{m_\text{sugar}RT}{M_\text{sugar} \cdot m_\text{water}} \Rightarrow M_\text{sugar} = \frac{m_\text{sugar}RT}{P \cdot m_\text{water}}\]

Use the given values to find the molecular weight of sugar

Now, plug in the given values and solve for \(M_\text{sugar}\): \[M_\text{sugar} = \frac{9.0 \: g \cdot 0.0821 \: \frac{L \cdot atm}{mol \cdot K} \cdot 300.15 \: K}{2.46 \: atm \cdot 0.5 \: kg}\] \[M_\text{sugar} = \frac{222.00045 \: g \cdot L \cdot atm}{1.23 \: mol \cdot kg}\] \[M_\text{sugar} = 180.49 \: g/mol\] The molecular weight of the sugar is approximately 180.49 g/mol.

Key concepts

Molality

Molality is a crucial concentration measure in chemistry, particularly when working with colligative properties such as osmotic pressure. Unlike molarity, which is affected by changes in temperature and pressure because it involves volume, molality is temperature and pressure independent because it is based on the mass of the solvent.
Molality (c) is defined as the number of moles of solute divided by the kilograms of solvent. The commonly used unit for molality is moles per kilogram (mol/kg).
When solving problems involving osmotic pressure, understanding molality is critical for setting up the correct equations. In the textbook problem, the molality is calculated by dividing the number of moles of sugar (solute) by the mass of water (solvent) in kilograms. However, since the problem does not provide the number of moles directly, you have to express molality in terms of known quantities—mass of sugar and molecular weight of sugar (which we're trying to find)—using the formula:

Molality using mass and molecular weight

\[c = \frac{m_{\text{sugar}}}{M_{\text{sugar}} \cdot m_{\text{water}}}\]
This understanding helps in correctly setting up the following steps for the calculation of molecular weight using the provided osmotic pressure.

Ideal Gas Constant

The ideal gas constant is a fundamental constant in the realm of physics and chemistry and plays an important role in equations describing the state of an ideal gas. In the osmotic pressure formula, the ideal gas constant is denoted by (R) and it appears in various units depending on the equation it is used in. For osmotic pressure calculations that involve molality, the constant is typically represented as 0.0821 L atm/mol K.
This constant provides a link between the physical properties of the gas and the units being used in calculations. It connects thermodynamic quantities such as temperature, with chemical quantities, namely moles, while also meshing nicely with the pressure and volume units.

Significance of ideality

In osmotic pressure calculations, sugar solutions can often be assumed to behave ideally, meaning they are dilute enough for the sugar molecules to act independently of each other, akin to ideal gas particles. This allows the use of the ideal gas constant in calculating osmotic pressure, acknowledging the solution's similarity to an ideal gas in terms of particle interaction and distribution.

Molecular Weight

Molecular weight (sometimes called molecular mass) represents the mass of one mole of a substance. It is a fundamental concept for chemists as it allows for the conversion between moles, which measure the quantity of molecules, and grams, which measure the weight. Molecular weight is expressed in grams per mole (g/mol).
Understanding the molecular weight is essential for many laboratory calculations, especially when preparing solutions of known concentration or when quantifying reactions. In the example problem we are discussing, knowing the molecular weight is the key to unlocking the amount of sugar required to achieve a certain osmotic pressure.

Application in osmotic pressure calculation

Once the osmotic pressure equation is rearranged to solve for molecular weight, the provided numerical values from the experiment (mass of sugar, temperature, osmotic pressure, and the mass of water) are used to calculate the molecular weight of sugar. The equation shows a proportional relationship between molecular weight and the known quantities. So, by solving for the molecular weight, we can understand the behavior of sugar in a solution at an osmotic level, a valuable insight for those studying solution chemistry or working with similar systems in a practical setting.