Q.Determine the amount of $CaCl_2$ ($i = 2.47$) dissolved in 2.5 litre of water such that its osmotic pressure is 0.75 atm at 27$^\circ$C.

The key idea is the van't Hoff equation for osmotic pressure, which relates osmotic pressure to the concentration of solute particles, accounting for dissociation.

1. First, convert the temperature to Kelvin: $T = 27^\circ\text{C} + 273 = 300 \text{ K}$.
2. Rearrange the osmotic pressure formula to solve for the number of moles ($n$), noting that $C = n/V$: $\Pi = i \frac{n}{V} RT \implies n = \frac{\Pi V}{iRT}$
3. Substitute the given values ($\Pi = 0.75 \text{ atm}$, $V = 2.5 \text{ L}$, $i = 2.47$, $R = 0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1}$, $T = 300 \text{ K}$): $n = \frac{0.75 \text{ atm} \times 2.5 \text{ L}}{2.47 \times 0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1} \times 300 \text{ K}}$ …

Osmotic pressure is a colligative property, meaning it depends only on the number of solute particles in a given volume of solution, not on their identity. When a solute is dissolved in a solvent, it lowers the solvent's chemical potential. If this solution is separated from the pure solvent by a semi-permeable membrane, solvent molecules will spontaneously move from the pure solvent side to the solution side to equalize the chemical potential. This movement is called osmosis.

Osmotic pressure ($\Pi$) is the external pressure that must be applied to the solution to stop the net flow of solvent across the semi-permeable membrane into the solution. For dilute solutions, osmotic pressure obeys a relation analogous to the ideal gas law — the van't Hoff equation:

In this problem, $CaCl_2$ is an electrolyte. When dissolved in water, it dissociates into ions:

Ideally, one mole of $CaCl_2$ would produce three moles of ions ($i=3$). However, the problem provides an experimental van't Hoff factor $i = 2.47$. This value is less than 3, indicating that some ion pairing occurs in the solution, reducing the effective number of particles. We must use the given $i = 2.47$.

Here's how we solve it step-by-step:

1. Identify the given values and the target:

   • Osmotic pressure, $\Pi = 0.75 \text{ atm}$
   • Volume of solution, $V = 2.5 \text{ L}$
   • Temperature, $T = 27^\circ\text{C}$
   • van't Hoff factor, $i = 2.47$
   • Gas constant, $R = 0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1}$
   • Target: mass of $CaCl_2$.

2. Convert temperature to Kelvin:

   The temperature in the osmotic pressure equation must be absolute:

   $T = 27 + 273 = 300 \text{ K}$

3. Rearrange the van't Hoff equation for the moles of solute.

   Since $C = \dfrac{n}{V}$, the equation $\Pi = iCRT$ becomes $\Pi = i\,\dfrac{n}{V}\,RT$, so:

4. Calculate the moles of $CaCl_2$ ($n$): $n = \frac{0.75 \text{ atm} \times 2.5 \text{ L}}{2.47 \times 0.0821 \text{ L atm mol}^{-1}\text{ K}^{-1} \times 300 \text{ K}}$ …

Method: Van’t Hoff Osmotic Pressure Equation

This method uses the relation between osmotic pressure, concentration, and the van’t Hoff factor to find the mass of solute.

Steps

Step 1: Write the van’t Hoff equation

Where:

• $\pi$ = osmotic pressure (atm)
• $i$ = van’t Hoff factor (given)
• $C$ = molar concentration (mol/L)
• $R$ = ideal gas constant = $0.0821 \ \text{L·atm·mol}^{-1}\text{K}^{-1}$
• $T$ = absolute temperature (K)

Step 2: Convert temperature to Kelvin

Step 3: Rearrange the equation to find molar concentration $C$

Substitute values:

Step 4: Calculate $C$

First compute denominator:

Now:

$C = \frac{0.75}{60.8361} \approx 0.01233 \ \text{mol/L}$ …

Here are the most common mistakes students make on this osmotic pressure / molar mass problem, and how to avoid each.

1. Forgetting the van’t Hoff factor ($i$)

Mistake: Using $\pi = CRT$ directly, without multiplying by $i$.

Why it happens: Students often treat non-electrolyte and electrolyte solutions the same way. Here, $CaCl_2$ dissociates, so the number of particles in solution is greater than the number of formula units.

How to avoid: Always check if the solute is ionic. The correct formula is:

2. Using the wrong value of $R$

Mistake: Using $R = 0.0821\,\text{L atm mol}^{-1}\text{K}^{-1}$ but forgetting to match units, or using $R = 8.314\,\text{J mol}^{-1}\text{K}^{-1}$ without converting pressure to Pa.

How to avoid: Since pressure is in atm and volume in litres, use $R = 0.0821\,\text{L atm mol}^{-1}\text{K}^{-1}$.

3. Not converting temperature to Kelvin

Mistake: Plugging in $T=27^\circ C$ directly.

How to avoid: $T(K) = 27+273 = 300\,\text{K}$.

4. Confusing molarity with moles

Mistake: Solving for $C$ (molarity) and then stopping, or using $C$ directly as moles.

How to avoid: After finding $C$, multiply by volume to get moles: $n = C\times V$. …