Problem 76 - An Introduction to Thermal Physics

Question

Seawater has a salinity of 3.5\%, meaning that if you boil away a kilogram of seawater, when you're finished you'll have 35 g of solids (mostly \(\mathrm{NaCl}\) ) left in the pot. When dissolved, sodium chloride dissociates into separate \(\mathrm{Na}^{+}\) and \(\mathrm{Cl}^{-}\) ions. (a) Calculate the osmotic pressure difference between seawater and fresh water. Assume for simplicity that all the dissolved salts in seawater are \(\mathrm{NaCl}\). (b) If you apply a pressure difference greater than the osmotic pressure to a solution separated from pure solvent by a semipermeable membrane, you get reverse osmosis: a flow of solvent out of the solution. This process can be used to desalinate seawater. Calculate the minimum work required to desalinate one liter of seawater. Discuss some reasons why the actual work required would be greater than the minimum.

Step-by-Step Solution

Answer

The osmotic pressure is 58.5 atm; minimum desalination work is about 5923 J for 1 L of seawater. Actual work is higher due to inefficiencies and additional dissolved substances.

Step 1: Determine the Molality of NaCl

First, we find out how many moles of NaCl are present in 1 kg of seawater. Given that seawater has a salinity of 3.5%, this means there are 35 g of NaCl in 1 kg of seawater. The molar mass of NaCl is 58.44 g/mol.Calculate the moles of NaCl:\[\text{moles of NaCl} = \frac{35 \text{ g}}{58.44 \text{ g/mol}} \approx 0.599 \text{ mol}\]This 0.599 mol of NaCl is present in 1 kg (1000 g) of seawater.

Step 2: Apply van 't Hoff's Factor

Since NaCl dissociates into two ions, we need to use the van 't Hoff factor, \(i\), which for NaCl is 2. Therefore, for osmotic pressure calculations, we consider twice the amount of solute particles.

Step 3: Calculate the Osmotic Pressure

The formula for osmotic pressure is given by:\[\Pi = i M R T\]Where \(\Pi\) is the osmotic pressure, \(M\) is the molarity (moles of solute per liter of solution), \(R\) is the ideal gas constant (0.0821 L·atm/mol·K), and \(T\) is the temperature in Kelvin.We have 0.599 mol of NaCl, considering the van 't Hoff factor, we use 1.198 mol (since \(i = 2\)). Convert the mass of seawater to liters assuming the density is close to that of water (1 kg = 1 L):\[M = \frac{1.198 \text{ mol}}{1 \text{ L}} = 1.198 \text{ M}\]Assuming a temperature of 25°C (298 K), plug in the values:\[\Pi = 2 \times 1.198 \times 0.0821 \times 298 \approx 58.5 \text{ atm}\]

Step 4: Calculate the Minimum Work for Desalination

The minimum work required for desalination, using reverse osmosis, is essentially the work needed to overcome the osmotic pressure per liter of water.The work, \(W\), is given by:\[W = \Pi \times V\]Where \(V\) is the volume of water in liters. For 1 liter of seawater:\[W = 58.5 \text{ atm} \times 1 \text{ L} \approx 58.5 \text{ L·atm} \approx 5923 \text{ J} \ (Converted: 1 \text{ L·atm} \approx 101.3 \text{ J})\]

Step 5: Discuss Why Actual Work is Greater

In real-world applications, the actual work required for desalination is greater than the calculated minimum due to factors such as the inefficiency of pumps and other mechanical equipment (friction, heat losses), the presence of other dissolved substances that raise the effective osmotic pressure, and the energy needed to maintain circulation and remove heat.

Key Concepts

Reverse Osmosis

Reverse osmosis is a fascinating process that helps separate solvent molecules from a solution through a semipermeable membrane. This is especially useful in applications like desalination, where the goal is to remove salt from seawater to produce safe drinking water. By applying pressure greater than the osmotic pressure of the solution, solvent molecules, like water, move from the salty solution to the freshwater side.
This effectively leaves behind the solutes, such as salt ions, thereby desalinating the seawater.

• Reverse osmosis requires precise pressure application to ensure efficiency.
• It is energy-intensive; hence, optimizing the process is crucial to achieving practical usage.

Innovations in reverse osmosis technologies focus on enhancing membrane durability and reducing energy consumption to make the process more economically feasible.

Desalination

Desalination is the process of removing salt and other minerals from seawater to produce fresh, potable water. This process is crucial in regions where freshwater is scarce.
By employing techniques like reverse osmosis, salts are effectively separated from water, reducing salinity levels to acceptable standards for human consumption and agriculture.

• Desalinated water helps alleviate shortages and provides a stable water source.
• However, desalination can be costly and requires a substantial amount of energy.

Efficient desalination practices aim to minimize costs and environmental impacts while maximizing water recovery rates. The continuous advancement of desalination technologies is vital for future water security.

Sodium Chloride Dissociation

When sodium chloride (NaCl), commonly known as table salt, dissolves in water, it undergoes dissociation. This means that the NaCl crystal lattice breaks down, and the individual sodium (Na⁺) and chloride (Cl⁻) ions are released into the solution.
This dissociation is essential for understanding the behavior of salt in solutions, especially regarding osmotic pressure calculations.

• The dissociation process increases the number of solute particles, influencing colligative properties.
• In seawater, primarily composed of NaCl, these free-moving ions significantly impact its osmotic pressure compared to freshwater.

Recognizing the behavior of sodium chloride in water helps in analyzing and utilizing various applications like water purification and industrial processes.

Van 't Hoff Factor

The Van 't Hoff factor (i) is a key concept in the study of solutions, particularly in the calculation of osmotic pressure, boiling point elevation, and freezing point depression. It indicates the number of particles a solute releases into the solution.
For NaCl, the Van 't Hoff factor is 2 because, upon dissociation, it produces two ions: one sodium ( Na⁺) ion and one chloride (Cl⁻) ion per formula unit.

• The Van 't Hoff factor is crucial for accurately determining properties like osmotic pressure.
• This factor allows us to account for the increased number of particles in a solution compared to the number of formula units dissolved.

Correctly applying this factor in calculations ensures that the physical properties of the solution are understood in a realistic context, guiding processes like desalination and chemical manufacturing.