Question

What is the osmotic pressure (in atm) of a \(1.57-M\) aqueous solution of urea \(\left[\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}\right]\) at \(27.0^{\circ} \mathrm{C} ?\)

Short answer

The osmotic pressure is approximately 38.66 atm.

Step-by-step solution

Understand the Formula for Osmotic Pressure

The formula for osmotic pressure is given by \( \Pi = iMRT \), where \( \Pi \) is the osmotic pressure, \( i \) is the van't Hoff factor, \( M \) is the molarity of the solution, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Identify Values and Constants

For urea, the van't Hoff factor \( i \) is 1 because urea does not dissociate in solution. Given molarity \( M = 1.57 \, M \), the temperature \( T = 27.0^{\circ} \mathrm{C} \) (which converts to \( 300.15 \, K \)), and the ideal gas constant \( R = 0.0821 \, \text{L atm/mol K} \).

Calculate Temperature in Kelvin

Convert the temperature from Celsius to Kelvin using the formula: \( T = 27.0 + 273.15 = 300.15 \, K \).

Apply Values to the Osmotic Pressure Formula

Substitute the values into the osmotic pressure formula: \( \Pi = 1 \times 1.57 \, M \times 0.0821 \, \text{L atm/mol K} \times 300.15 \, K \).

Perform the Calculation

Simplify the expression: \( \Pi = 1.57 \times 0.0821 \times 300.15 = 38.66 \, \text{atm} \).

State the Final Answer

The osmotic pressure of the solution is approximately \( 38.66 \, \text{atm} \).

Key concepts

Van't Hoff Factor

The van't Hoff factor, abbreviated as \( i \), is a crucial element in calculating colligative properties such as osmotic pressure. It represents the number of particles into which a solute dissociates in solution. For example, electrolytes like sodium chloride (NaCl) can dissociate into two ions – sodium \((\text{Na}^+)\) and chloride \((\text{Cl}^-)\), giving an \( i \) value of 2.
Non-electrolytes like urea do not dissociate in solution, thus their van't Hoff factor is 1. This simplicity makes calculations straightforward. The closer the actual \( i \) value is to the theoretical, the more ideal the solution behaves. Calculating with \( i \) helps us correct for real-world non-ideal behaviors in solutions.

Ideal Gas Constant

The ideal gas constant \( R \) is a fundamental constant used in many important equations in chemistry, including the ideal gas law \( PV = nRT \), and osmotic pressure equation \( \Pi = iMRT \). It links pressure, volume, temperature, and the number of moles of gas in a sample.
Its value is \( 0.0821 \, \text{L atm/mol K} \) when dealing with pressure in atmospheres, volume in liters, and temperature in Kelvin. It appears in various forms depending on the units used. Understanding \( R \) is essential for analyzing how gases behave under different conditions, providing a bridge between different states and transformations in physical chemistry.

Temperature Conversion

Temperature conversion, especially from Celsius to Kelvin, is a fundamental skill in chemistry. The Kelvin scale is the SI unit for temperature and is necessary for consistency in equations like those for osmotic pressure.
To convert Celsius to Kelvin, simply add \( 273.15 \). For instance, \( 27.0^{\circ} \mathrm{C} \) becomes \( 300.15 \, K \). This conversion ensures that temperature is always in absolute units, preventing mathematical inconsistencies and errors due to zero or negative temperatures in calculations. Understanding and using Kelvin in thermodynamic equations is vital for accurate scientific measurement and analysis.

Molarity

Molarity \( M \) is a measure of concentration in solution, expressed in moles of solute per liter of solution. It is crucial for calculating osmotic pressure, among other colligative properties. A molarity of \( 1.57 \) \( M \) indicates there are \( 1.57 \) moles of urea per liter of solution.
This unit helps chemists determine how solutions interact and transform during reactions. One of the benefits of working with molarity is its straightforward use in stoichiometry and preparation of solutions, allowing easy prediction and control over chemical processes. Molarity is a key concept in understanding solutions and reactions in chemistry.