Question

A water desalination plant is set up near a salt marsh containing water that is \(0.10 \mathrm{M} \mathrm{NaCl}\). Calculate the minimum pressure that must be applied at \(20 .{ }^{\circ} \mathrm{C}\) to purify the water by reverse osmosis. Assume \(\mathrm{NaCl}\) is completely dissociated.

Short answer

The minimum pressure that must be applied at \(20 .{ }^{\circ} \mathrm{C}\) to purify the water by reverse osmosis is approximately \(4.8\mathrm{atm}\).

Step-by-step solution

Write down the given information

We are given the concentration of the saltwater solution as \(0.10 \mathrm{M} \mathrm{NaCl}\). We know that NaCl is completely dissociated in the solution. The temperature is provided as \({20 .{ }}^{\circ} \mathrm{C}\).

Convert the temperature to Kelvin

In order to use the osmotic pressure formula, we need to convert the temperature from Celsius to Kelvin. We know that the relationship between Celsius and Kelvin is: K = °C + 273.15. Therefore, we have: \[T = 20 ^{\circ} \mathrm{C} + 273.15 = 293.15 \mathrm{K}\]

Identify the gas constant

The gas constant (R) is a constant value used in calculations involving ideal gases. In this case, since we are dealing with osmotic pressure, the value of the gas constant should be in units of L atm/mol K. The value of the gas constant R is given as: \[R=0.0821\frac{\mathrm{L\ atm}}{\mathrm{mol\ K}}\]

Identify the concentration of the dissolved particles

We are given the concentration of NaCl as \(0.10 \mathrm{M}\). Since NaCl completely dissociates in the solution, we must consider the concentration of both Na+ and Cl- ions. For every mole of NaCl, there is one mole of Na+ and one mole of Cl-. Thus, the total concentration of dissolved particles is: \[C_\text{total} = C_\text{Na+} + C_\text{Cl-} = 2 \times 0.10 \mathrm{M} = 0.20\mathrm{M}\]

Calculate the osmotic pressure

Osmotic pressure (π) can be calculated using the following formula: \[\pi = C_\text{total} \times R \times T\] Plug in the values of concentration, temperature and gas constant into the formula: \[\pi = 0.20\mathrm{M} \times 0.0821\frac{\mathrm{L\ atm}}{\mathrm{mol\ K}} \times 293.15\mathrm{K}\] Calculate the osmotic pressure: \[\pi \approx 4.8\mathrm{atm}\] The minimum pressure that must be applied at \(20 .{ }^{\circ} \mathrm{C}\) to purify the water by reverse osmosis is approximately \(4.8\mathrm{atm}\).

Key concepts

Osmotic Pressure Calculation

When discussing reverse osmosis in chemistry, understanding how to calculate osmotic pressure is crucial. Osmotic pressure is the pressure required to prevent water from flowing across a semipermeable membrane due to osmosis. It's influenced by factors such as solute concentration and temperature.

To calculate osmotic pressure, we use the formula:
\[\pi = C_{\text{total}} \times R \times T\] where \(\pi\) is the osmotic pressure, \(C_{\text{total}}\) is the molar concentration of dissolved particles, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. It's important to remember that the concentration used in the formula is the total concentration of all particles in solution, which leads us to the concept of dissociation.

Dissociation of NaCl

In the context of osmotic pressure, dissociation refers to how a compound like NaCl (sodium chloride) separates into its ions when dissolved in water. This is critical because the osmotic pressure depends on the total number of dissolved particles, not just the original concentration of the compound.

When NaCl dissolves, it separates into sodium ions (Na+) and chloride ions (Cl-). Each unit of NaCl produces one Na+ and one Cl-, effectively doubling the number of particles and, subsequently, the osmotic pressure. Therefore, you must account for complete dissociation when calculating osmotic pressure; otherwise, you would underestimate it.

Gas Constant R Value

The gas constant, denoted as \(R\), is a fundamental constant in chemistry, prominently featured in the ideal gas law among other equations. For calculations involving osmotic pressure, the value of \(R\) is typically \(0.0821 \frac{\text{L atm}}{\text{mol K}}\). It is important to use the correct units for \(R\) to ensure that the osmotic pressure is calculated correctly. The unit of pressure in the result will depend on the units used for the gas constant. Always confirm that the units for \(R\) align with the other values in your calculation.

Conversion of Celsius to Kelvin

Temperature plays a significant role in calculating osmotic pressure and must be expressed in Kelvin for the formula to work correctly. Kelvin is the SI unit for thermodynamic temperature and is used in scientific equations to avoid negative values, since the Kelvin scale starts at absolute zero.

To convert Celsius to Kelvin, you add 273.15 to the Celsius temperature:
\[T_{\text{K}} = T_{\text{C}} + 273.15\] where \(T_{\text{K}}\) is the temperature in Kelvin and \(T_{\text{C}}\) is the temperature in Celsius. This conversion is straightforward but essential to ensure accurate osmotic pressure calculations.