Question

A solution contains \(8.92 \mathrm{~g}\) of \(\mathrm{KBr}\) in \(500.0 \mathrm{~mL}\) of solution and has an osmotic pressure of 6.97 atm at \(25^{\circ} \mathrm{C}\). Calculate the van't Hoff factor ( \(i\) ) for \(\mathrm{KBr}\) at this concentration.

Short answer

After calculating the molarity of the solution and rearranging van't Hoff's law, the calculated van't Hoff factor (\(i\)) for KBr at this concentration is found to be the ratio of the osmotic pressure to the product of the molarity, gas constant, and temperature in Kelvin.

Step-by-step solution

Convert Temperature to Kelvin

To use the formula involving osmotic pressure, we need the absolute temperature in kelvin. Convert the given temperature from Celsius to Kelvin using the formula: Kelvin = Celsius + 273.15. So, T = 25 + 273.15 = 298.15 K.

Calculate Molarity

Molarity (M) is moles of solute per liter of solution. Calculate the moles of KBr by dividing the mass by its molar mass (KBr = 39.1 g/mol for K + 79.9 g/mol for Br = 119 g/mol). Then divide the moles of KBr by the volume of the solution in liters to find the molarity. M = (8.92 g / 119 g/mol) / (500 mL / 1000 mL/L).

Apply van't Hoff's Law

Use the van't Hoff's law form \(\(\Pi = i M R T\)\) where \(\Pi\) is the osmotic pressure, \(i\) is the van't Hoff factor, \(M\) is the molarity, \(R\) is the ideal gas constant (0.0821 L·atm/(K·mol)), and \(T\) is the temperature in Kelvin. Solve for \(i\) by rearranging the equation to \(i = \frac{\Pi}{M R T}\).

Calculate the van't Hoff Factor

Substitute the values of osmotic pressure (6.97 atm), molarity (calculated in Step 2), the ideal gas constant, and the temperature in Kelvin into the rearranged van't Hoff's law to solve for \(i\).

Key concepts

Understanding Osmotic Pressure

Osmotic pressure is a critical concept in chemistry, particularly in the study of solutions. It is the pressure required to stop the flow of a solvent through a semipermeable membrane into a solution. When dealing with osmotic pressure, we often refer to van't Hoff's law which relates osmotic pressure with the molarity of the solution, the temperature, and the van't Hoff factor of the solute.

To simplify, osmotic pressure increases as the number of dissolved particles in the solution increases. That's where the van't Hoff factor (\(i\)) comes in; it accounts for the number of particles into which a substance dissociates in solution. For example, NaCl dissociates into Na\textsuperscript{+} and Cl\textsuperscript{-}, typically giving it a van't Hoff factor of 2. However, the actual factor can vary due to ion pairing or other interactions within the solution.

Molarity Calculation Demystified

Molarity is a measure of the concentration of a solution, representing the moles of a solute per liter of solution. It's often denoted by 'M' and is critical for quantitative analysis in chemistry.

To calculate molarity, you'll need the mass of the solute (in grams), its molar mass (grams per mole, which can be found using the periodic table), and the volume of the solution (in liters). The formula is simple: \(M = \frac{\text{moles of solute}}{\text{liters of solution}}\).
Remember to convert volumes from milliliters to liters when necessary, since molarity calculations require liters for an accurate readout. This conversion is straightforward: 1000 milliliters equal 1 liter.

Temperature Conversion, A Key to Understanding

Temperature plays a vital role in various chemical processes and thus it is crucial to master the conversion between different temperature scales. In chemistry, especially when working with gas laws and calculating osmotic pressure, temperatures need to be expressed in the Kelvin scale.

The Kelvin scale is an absolute temperature scale, which means it starts from absolute zero, the lowest theoretical temperature possible. To convert from Celsius to Kelvin, which is a common conversion in chemistry, you can use the formula: \(K = {}^\circ C + 273.15\).
This conversion is essential because the Kelvin scale allows for proper insertion into various gas law equations, including calculations for osmotic pressure.

Gas Laws in Chemistry Explained

Gas laws in chemistry describe how gases behave under varying conditions of temperature, volume, and pressure. They are fundamental to understanding how chemicals react and behave in different situations.
Key gas laws include Boyle's Law, which states that pressure and volume are inversely related; Charles's Law, which links volume and temperature; and Avogadro's Law, connecting volume and moles of gas.
Then there's the Ideal Gas Law, which is a combination of all these individual laws, formulated as \(PV = nRT\), where P is pressure, V is volume, n is moles, R is the ideal gas constant, and T is temperature in Kelvin.
This law is at the heart of the relationship between osmotic pressure and molarity in solutions, broadening its relevance beyond just gases to include solutions and their behaviors.