Consider separate solutions of `0.500 M C_(2)H_(5)OH(aq)`,`0.100 M Mg_(3)(PO_(4))(aq)`,`0.250 M KBr(aq)`, and `0.125 M Na_(3)PO_(4)(aq)` at `25^(@)C`. Which statement is true about these solutions, assuming all salts to be strong electrolytes?

Options

A. `0.125 M Na_(3)PO_(4)(aq)` has the highest osmotic pressure.
B. `0.500 M C_(2)H_(5)OH` has the highest osmotic pressure.
C. They all have the same osmotic pressure.
D. `0.100 M Mg_(3)(PO_(4))_(2)(aq)` has the highest osmotic pressure.

Explanation

To determine which statement is true about the given solutions, we need to calculate the osmotic pressure for each solution. The osmotic pressure (π) can be calculated using the formula:

\[ \pi = iCRT \]

Where:
- \(i\) = van 't Hoff factor (the number of particles the solute dissociates into)
- \(C\) = molarity of the solution
- \(R\) = ideal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature in Kelvin (25°C = 298 K)

Let's analyze each solution:

1. **For \(C_2H_5OH(aq)\) (Ethanol)**:
- Ethanol is a non-electrolyte, so \(i = 1\).
- \(C = 0.500 \, M\)
- Therefore, \(\pi = 1 \times 0.500 \times 0.0821 \times 298\)

2. **For \(Mg_3(PO_4)(aq)\)**:
- This salt dissociates into 5 ions: \(3 Mg^{2+}\) and \(1 PO_4^{3-}\), so \(i = 5\).
- \(C = 0.100 \, M\)
- Therefore, \(\pi = 5 \times 0.100 \times 0.0821 \times 298\)

3. **For \(KBr(aq)\)**:
- This salt dissociates into 2 ions: \(K^+\) and \(Br^-\), so \(i = 2\).
- \(C = 0.250 \, M\)
- Therefore, \(\pi = 2 \times 0.250 \times 0.0821 \times 298\)

4. **For \(Na_3PO_4(aq)\)**:
- This salt dissociates into 4 ions: \(3 Na^+\) and \(1 PO_4^{3-}\), so \(i = 4\).
- \(C = 0.125 \, M\)
- Therefore, \(\pi = 4 \times 0.125 \times 0.0821 \times 298\)

Now, let's calculate the osmotic pressures for each solution:

### Step-by-Step Calculations:

1. **Ethanol**:
\[ \pi = 1 \times 0.500 \times 0.0821 \times 298 = 12.21 \, atm \]

2. **Magnesium Phosphate**:
\[ \pi = 5 \times 0.100 \times 0.0821 \times 298 = 1.24 \, atm \]

3. **Potassium Bromide**:
\[ \pi = 2 \times 0.250 \times 0.0821 \times 298 = 1.54 \, atm \]

4. **Sodium Phosphate**:
\[ \pi = 4 \times 0.125 \times 0.0821 \times 298 = 1.23 \, atm \]

### Summary of Osmotic Pressures:
- Ethanol: \(12.21 \, atm\)
- Magnesium Phosphate: \(1.24 \, atm\)
- Potassium Bromide: \(1.54 \, atm\)
- Sodium Phosphate: \(1.23 \, atm\)

### Conclusion:
From the calculations, we can see that the solution with the highest osmotic pressure is that of ethanol, followed by potassium bromide, magnesium phosphate, and sodium phosphate.