Question

Consider two solutions at a certain temperature. Solution X has a nonelectrolyte as a solute and an osmotic pressure of 1.8 atm. Solution \(Y\) also has a nonelectrolyte as a solute and an osmotic pressure of 4.2 atm. What is the osmotic pressure of a solution made up of equal volumes of solutions \(\mathrm{X}\) and \(\mathrm{Y}\) at the same temperature? Assume that the volumes are additive.

Short answer

Answer: The osmotic pressure of the resulting solution when equal volumes of solution X and Y are mixed is 9 atm.

Step-by-step solution

Recall the osmotic pressure formula

The osmotic pressure formula is given by: \(\Pi = cRT\), where \(\Pi\) is the osmotic pressure, \(c\) is the molar concentration of the solute, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Determine the initial molar concentrations of solutions X and Y

We are given osmotic pressures for both solutions and their temperature. Let's assume \(c_X\) is the molar concentration of solution X, and \(c_Y\) is the molar concentration of solution Y. Therefore, we can write the osmotic pressure formula for both solutions: \(\Pi_X = c_XRT\) and \(\Pi_Y = c_YRT\) Since R and T are constants, we can determine the ratio of the molar concentrations: \(\frac{c_X}{c_Y} = \frac{\Pi_X}{\Pi_Y}\) Now, use the given osmotic pressures to find the ratio of molar concentrations: \(\frac{c_X}{c_Y} = \frac{1.8\,\text{atm}}{4.2\,\text{atm}} = \frac{3}{7}\)

Find the total molar concentration in the resulting mixture

As Solution X and Y are mixed in equal volumes V, the total volume of the resulting solution becomes 2V. To find the total molar concentration in the resulting mixture, we need to account for the moles from both solutions: \(c_{total} = \frac{c_X \times V + c_Y \times V}{2V}\) Substituting the ratio of molar concentrations from step 2 and simplifying the equation, we get: \(c_{total} = \frac{1}{2}(c_X + c_Y) = \frac{1}{2}(c_X + 7c_X) = 5c_X\)

Calculate the osmotic pressure of the resulting mixture

Finally, use the osmotic pressure formula for the resulting mixture with the total molar concentration we determined in step 3: \(\Pi_{total} = c_{total}RT = 5c_XRT\) Since we have the osmotic pressure for solution X, we can use it to find the osmotic pressure of the resulting mixture: \(\Pi_{total} = 5 \times 1.8\,\text{atm} = 9\,\text{atm}\) Thus, the osmotic pressure of the resulting solution when equal volumes of solution X and Y are mixed is 9 atm.