Question

A water desalination plant is set up near a salt marsh containing water that is \(0.10 M\) 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

Based on the osmotic pressure formula, the minimum pressure required to purify the water is \(π = \frac{nRT}{V}\). With the given concentration of NaCl (\(0.10 M\)), temperature (\(20^{\circ}C\)), and assuming NaCl is completely dissociated, we can calculate the minimum pressure to be approximately \(4.92 atm\).

Step-by-step solution

Recall the formula for osmotic pressure

We will use the osmotic pressure formula to find out the minimum pressure required to purify the water. The formula for osmotic pressure is: \(π = nRT/V\) Where: π = osmotic pressure n = number of moles of solute R = gas constant (0.08206 L atm/mol K) T = temperature in Kelvin V = volume in liters

Key concepts

Understanding Osmosis

Osmosis is a natural process involving the movement of solvent molecules through a semipermeable membrane from a region of lower solute concentration to a region of higher solute concentration. In simpler terms, imagine water molecules traveling through a barrier that only lets those molecules through. They naturally move towards an area with more solute, like salt, to achieve equilibrium.
In the context of water purification, osmosis is the basis for reverse osmosis. Normally, water would naturally flow into the saltwater because it is more concentrated. Reverse osmosis uses pressure to do the opposite, forcing water out of saltwater and leaving salts behind.

What is Osmotic Pressure?

Osmotic pressure is the pressure required to stop this natural flow of water across the semipermeable membrane. Think of it as a force that's used to counteract osmosis.
This pressure can be calculated using the formula \(π = iMRT\), where:

• \(π\) is the osmotic pressure
• \(i\) is the van't Hoff factor (which equals 2 for NaCl as it dissociates into two ions, Na⁺ and Cl⁻)
• \(M\) is the molarity of the solution
• \(R\) is the gas constant
• \(T\) is the temperature in Kelvin

In our exercise, to reverse osmosis in desalination practice, the applied pressure must be greater than the osmotic pressure of the saline solution.

Desalination and Its Importance

Desalination is the process of removing salt and other impurities from seawater or brackish water to make it suitable for human consumption or irrigation. It's a crucial technology in areas with limited freshwater resources.
Reverse osmosis is a common method for desalination, where water is forced through a semipermeable membrane at high pressure. This prevents the salt and other impurities from passing through, leaving clean water on the other side.

• Environmental impact and economic viability remain challenges.
• Innovation continues to make this process more efficient and accessible.

Reverse osmosis contributes significantly to potable water resources in arid regions.

Sodium Chloride Dissociation

Sodium chloride, or common table salt, consists of sodium and chloride ions. When dissolved in water, NaCl dissociates completely into its constituent ions:

• \(NaCl \rightarrow Na^+ + Cl^-\)

This dissociation is key when calculating osmotic pressure because it increases the number of solute particles in the solution. The dissociation results in a greater osmotic pressure compared to a nondissociating solute of the same concentration.
Complete dissociation is assumed in our exercise, which means for every mole of NaCl, there are two moles of ions formed. This factor greatly influences the reverse osmosis process.