A \(0.010 \mathrm{~g}\) sample of \(\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{SO}_{4}\right) \mathrm{Cl}\) is dissolved in \(25.0 \mathrm{~mL}\) of water and the osmotic pressure of the solution is \(59.1\) torr at \(25^{\circ} \mathrm{C}\). How many moles of ions are produced per mole of compound? (a) 1 (b) 4 (c) 2 (d) 3

Understanding the van't Hoff factor formula is crucial when studying solutions and their properties. This formula is key for relating the number of particles in a solution to the observed colligative properties, such as osmotic pressure. The van't Hoff factor, denoted as \( i \), represents the number of particles a compound dissociates into when it is dissolved in a solvent. In the case of our osmotic pressure calculation, the formula is given by:
\( \pi = i M R T \)
where \( \pi \) is the osmotic pressure, \( M \) is the molarity of the solution, \( R \) is the ideal gas constant and \( T \) is the temperature in Kelvin.

To compute the osmotic pressure, one must know the dissociation state of the solute. For example, a salt like NaCl dissociates into two ions, Na⁺ and Cl⁻, meaning \( i = 2 \). In our exercise, we need to determine \( i \) by using the observed osmotic pressure and other calculated values. The solution will reveal the number of particles produced by the compound when dissolved.

Molarity is a term used to describe the concentration of a solution, often represented by the letter \( M \). It’s defined as the number of moles of solute per liter of solution. The formula to calculate molarity is:
\( Molarity = \frac{mass \, (g)}{molecular \, weight \, (g/mol) \times volume \, (L)} \)
When calculating molarity, it’s important to convert all your units properly. Mass must be in grams, volume in liters, and molecular weight in grams per mole. In our example, the molar concentration of the chromium complex compound is found using the mass of the sample and the volume of water it’s dissolved in. This step is paramount in determining the osmotic pressure using the van't Hoff formula, as it provides the value of \( M \) in that equation.

The concept of 'ions in solution' refers to the charged particles that are released when an ionic compound dissolves in a solvent like water. These ions are crucial for conducting electricity and participating in chemical reactions within the solution. The number of ions produced can significantly influence the solution’s osmotic pressure. For example, when a salt dissolves and dissociates completely, each molecule forms multiple ions. Our exercise involves a complex ion, \(\mathrm{Cr}\left(\mathrm{NH}_3\right)_4\left(\mathrm{SO}_4\right)\mathrm{Cl}\), which will dissociate into different ions in the solution. Determining how many ions are produced per mole of the dissolved compound is essential to understand the colligative properties like freezing point depression, boiling point elevation, and in this case, osmotic pressure.

Temperature conversion is a fundamental skill in chemistry because many calculations require temperature to be in a specific unit. Most commonly, scientific calculations demand temperatures in Kelvin, while many everyday situations use Celsius or Fahrenheit.
For conversion between Celsius and Kelvin, which directly relates to our exercise, the formula is:
\( T(K) = T(^\text{\textdegree} C) + 273.15 \)
This implies that zero on the Celsius scale (the freezing point of water) corresponds to 273.15 K. In the discussed exercise, the temperature given in Celsius must be converted to Kelvin to correctly apply the van't Hoff equation for osmotic pressure. Remember, when converting temperature for gas-related equations, Kelvin is always the unit of choice due to its absolute nature.