\(\mathrm{V}\) litres of a solution contains \(\mathrm{m}_{2} \mathrm{~g}\) of non volatile solute of molecular mass \(\mathrm{M}_{2}\). Which of the following can be used to calculate the molecular mass of solute in terms of osmotic pressure? (a) \(\frac{\mathrm{M}_{2}}{\mathrm{~V}}=\frac{\mathrm{m}_{2}}{\pi} \mathrm{VRT}\) (b) \(\frac{\mathrm{M}_{2}}{\mathrm{~V}}=\frac{\mathrm{m}_{2}}{\pi} \mathrm{RT}\) (c) \(\mathrm{M}_{2}=\frac{\mathrm{m}_{2}}{\mathrm{~V}} \pi \mathrm{R} \mathrm{T}\) (d) \(\mathrm{M}_{2}=\frac{\mathrm{m}_{2}}{\mathrm{~V}} \frac{\pi}{\mathrm{RT}}\)

Osmotic pressure is a fundamental concept in chemistry, closely related to the properties of solutions. It is defined as the pressure required to prevent osmosis, which is the movement of solvent molecules through a semipermeable membrane from a region of lower solute concentration to one of higher concentration. The formula that we use to quantify this pressure in a solution is given by \( \pi = CRT \), where \( \pi \) represents osmotic pressure, \( C \) is the concentration of the solute in moles per liter, \( R \) is the ideal gas constant (0.0821 L atm mol\(^{-1}\) K\(^{-1}\)), and \( T \) is the absolute temperature in Kelvin.

Understanding the relationships encapsulated in this formula makes it possible to derive other important quantities about a solution, such as molecular masses of solutes, by rearranging and manipulating the terms appropriately. It's critical to keep in mind the conditions under which this relationship holds, such as ideal behavior and no volume change on mixing, which are assumed for simplicity in most introductory chemistry contexts.

The utility of this formula lies in its adoption of the gas laws to systems that are not gaseous. Since osmotic pressure is akin to the pressure exerted by a gas, it can be modeled in a similar way, allowing us to apply concepts that are well understood from the behavior of gases to the behavior of solutions.

Molecular mass is an important characteristic of a solute, reflecting the mass of one mole of a substance. It is particularly significant when studying solutions, as it relates to the number of moles of solute present in a given amount of solvent. Using the osmotic pressure formula, it is possible to calculate the molecular mass of a non-volatile solute once the osmotic pressure of the solution is known.

The process begins with the fundamental equation \( \pi = CRT \). Since the concentration \( C \) can be expressed as the number of moles \( n \) divided by the volume in liters \( V \) (so \( C = \frac{n}{V} \)), and the number of moles can be calculated by dividing the mass of the solute \( m_2 \) by the molecular mass \( M_2 \) (so \( n = \frac{m_2}{M_2} \)), we can combine these relationships to express \( M_2 \) in terms of the known quantities and the osmotic pressure:
\[ M_2 = \frac{m_2 RT}{\pi V} \]
This calculation is a vital tool for chemists, biologists, and many other scientists, as it permits the determination of molecular masses without the need for sophisticated equipment or direct measurement. For students learning about solutions, mastering this calculation not only helps in solving textbook problems but also lays groundwork for understanding more complex concepts in solution chemistry.

Solution concentration is a measure of the amount of solute present in a solution compared to the volume of solvent. It plays a crucial role in how solutions behave, influencing properties like boiling point elevation, freezing point depression, and indeed osmotic pressure. It’s common to describe concentration in terms of molarity, which is the number of moles of solute per liter of solution.

When working with osmotic pressure, understanding concentration becomes especially significant, as the formula \( \pi = CRT \) directly incorporates concentration. Thus, accurately calculating and understanding molarity is essential for students to predict and explain the behavior of solutions. Furthermore, concentration calculations connect to numerous practical applications, such as preparing chemical reagents for laboratory experiments or administering medication dosages in pharmaceutical formulations.

To conclude, it’s vital to always ensure the units are consistent when performing calculations involving solution concentration. Failing to do so can lead to errors in the calculation of properties such as osmotic pressure or the molecular mass of a solute, as observed in the exercise. Through repeated practice, students will develop a strong grasp of how to work with concentrations, becoming adept at manipulating and applying these various formulae to solve complex problems in solution chemistry.