Animated Solution for Chemistry - Solutions: The osmotic pressure of a solution of NaCl is $0.10\text{ atm}$ and that of a glucose solution is $0.20\text{ atm}$. The osmotic pressure of a solution formed by mixing $1\text{ L}$ of the sodium chloride solution with $2\text{ L}$ of the glucose solution is $x\times 10^{-3}\text{ atm}$. $x$ is ......... (nearest integer).

Imagine you are standing in a laboratory with two separate beakers in front of you. In the first beaker, you have $1\text{ L}$ of a sodium chloride (NaCl) solution exerting an osmotic pressure of $0.10\text{ atm}$. In the second beaker, there is $2\text{ L}$ of a glucose solution with an osmotic pressure of $0.20\text{ atm}$. Your task is to mix these two solutions and determine the osmotic pressure of the final mixture.

This is a classic problem of colligative properties, specifically dealing with the mixing of non-reacting solutions. Let's break down the physics and math behind this beautiful setup.

To find the final osmotic pressure, we cannot simply add the individual pressures. Osmotic pressure depends on the total number of solute particles in a given volume. Therefore, we need to find the exact number of moles of particles each solution contributes to the final mixture.

The master equation for osmotic pressure is given by:

Here, $C$ is the molar concentration, which is the number of moles ($n$) divided by the volume ($V$). We can rewrite the equation to solve for the number of moles:

Let's analyze the first solution. Sodium chloride is a strong electrolyte. When dissolved in water, it completely dissociates into two ions: ${\text{Na}}^{+}$ and ${\text{Cl}}^{-}$. This means its van't Hoff factor ($i$) is $2$. Substituting the given values into our rearranged formula, we get the moles of NaCl:

Now, let's look at the second solution. Glucose is a non-electrolyte; it does not dissociate into ions in water. Therefore, its van't Hoff factor ($i$) is $1$. Calculating the moles of glucose:

When we mix the two solutions, the total volume becomes the sum of the individual volumes:

The total osmotic pressure of the mixture will depend on the total effective moles of particles. The effective moles are calculated by multiplying the actual moles of each solute by its respective van't Hoff factor. The formula for the total osmotic pressure is:

Let's substitute the values we found into this final equation:

Notice how the $RT$ term elegantly factors out and cancels from the numerator and denominator. This is why we didn't need the exact temperature or the gas constant! Simplifying the expression:

Dividing $0.50$ by $3$ gives us $\mathrm{0.1666...}\text{ atm}$. The question asks for the answer in the format of $x\times 10^{-3}\text{ atm}$. Let's convert our result:

Rounding this to the nearest integer, we get $167$. Therefore, the value of $x$ is $167$.

This problem beautifully demonstrates how colligative properties depend strictly on the number of particles. Always remember to account for the van't Hoff factor when dealing with electrolytes!