Problem 63 - General Chemistry: Principles and Modern Applications

Question

In what volume of water must \(1 \mathrm{mol}\) of a nonelectrolyte be dissolved if the solution is to have an osmotic pressure of 1 atm at 273 K? Which of the gas laws does this result resemble?

Step-by-Step Solution

Answer

The solution requires a volume of 22.4073 L of water. The result resembles the Ideal Gas Law.

Step 1: Reminder of the Osmotic Pressure Formula

Osmotic pressure (\(π\)) can be calculated using the formula: \(π = nRT/V\), where n is the number of moles, R is the ideal gas constant (0.0821 L.atm/mol.K), T is the temperature in Kelvin, and V is the volume.

Step 1: Substituting the Known Values

Substituting the known values into the equation: \(1 = 1*0.0821*273/V\). We need to solve this equation for V (volume).

Step 2: Solve for V (Volume)

To solve for V, we can apply algebraic operations. Multiplying the variables on the right side of the equation gives: 1 = 22.4073/V. Then, cross-multiplying yields V = 22.4073 L.

Step 3: Examine the Result and Gas Laws

Looking at the result, it can be seen that this resembles the ideal gas law equation \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. In the osmotic pressure equation, the osmotic pressure (π) stands as equivalent to the pressure (P) in the ideal gas law.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a critical concept in understanding the behavior of gases under various conditions. It is expressed as \(PV = nRT\), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume occupied by the gas.
• \(n\) is the number of moles of the gas.
• \(R\) is the ideal gas constant, approximately 0.0821 L.atm/mol.K.
• \(T\) is the temperature in Kelvin.

The equation shows the relationship between these variables, explaining how changing one affects the others. In the context of osmotic pressure, this law is analogous as it also relates pressure, volume, and temperature with the number of moles in a given substance. Remember, the Ideal Gas Law is an approximation for ideal cases where gases are assumed to have no interactions between molecules and occupy no volume themselves.
It can be directly applied to situations like our exercise, where pressure is related to the number of dissolved solute particles in a solution, assuming ideal solution conditions.

Nonelectrolyte Solution

A nonelectrolyte solution is one where the solute dissolved does not dissociate into ions. This means that the solute molecules remain intact in the solution. Common nonelectrolytes include sugar and urea. When dissolved, they do not conduct electricity because no charge carriers are produced.
For the osmotic pressure problem presented, it's crucial to consider the solution as a nonelectrolyte since the osmotic pressure equations are derived for solutes that do not form ions. This distinguishes nonelectrolytes from electrolytes, which dissociate into ions and produce a larger effect on colligative properties like osmotic pressure. Hence, when calculating osmotic pressure for nonelectrolyte solutions, the equation remains simple and direct, unlike electrolyte solutions where additional factors must be considered.

Van 't Hoff Equation

Derived from the ideal gas law, the Van 't Hoff equation for osmotic pressure quantifies the osmotic pressure \(π\) of a solution:

• \(π = \frac{nRT}{V}\)

Here:

• \(π\) is the osmotic pressure, analogous to the pressure \(P\) in the ideal gas law.
• \(n\) is the number of moles of solute.
• \(R\) is the ideal gas constant (0.0821 L.atm/mol.K).
• \(T\) is the temperature in Kelvin.
• \(V\) is the volume of the solvent in liters.

The Van 't Hoff equation aids in determining the osmotic pressure, assuming the solution behaves ideally. This technique is very useful in predicting how solutions will behave in biological and chemical contexts, where controlling the pressure differences across membranes is crucial.
Importantly, this equation reveals that osmotic pressure is directly proportional to both the temperature and the concentration of solute particles, as long as the solution remains dilute. This relationship helps underscore the similarity between osmotic processes and gaseous behavior described by the ideal gas law.