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

The minimum pressure that must be applied to purify the water by reverse osmosis at 20°C with a 0.10 M NaCl solution is approximately \(4.79 \, \mathrm{atm}\).

Step-by-step solution

Write down the reverse osmosis equation

The equation for calculating the minimum pressure in reverse osmosis is given by: \(P_{min} = i \cdot c \cdot R \cdot T\) Where: \(P_{min}\) = minimum pressure \(i\) = van't Hoff factor for NaCl (dissociation factor for electrolytes) \(c\) = molar concentration (in moles/liter) \(R\) = Ideal gas constant (0.0821 L atm/mol K) \(T\) = temperature in Kelvin

Calculate the van't Hoff factor for NaCl

The van't Hoff factor (i) for NaCl is calculated based on the degree to which it dissociates in solution. Since NaCl completely dissociates into its ions (\(\mathrm{Na}^+\) and \(\mathrm{Cl}^-\)) in the solution, the van't Hoff factor is 2, as there are two ions produced for each salt molecule dissolved.

Convert the temperature to Kelvin

To convert the given temperature from Celsius to Kelvin, we add 273.15 to the given value: \(T_{K} = 20^\circ \mathrm{C} + 273.15 = 293.15 \, \mathrm{K}\)

Substitute known values into the reverse osmosis equation

Now, we will substitute the known values into the reverse osmosis equation: \(P_{min} = i \cdot c \cdot R \cdot T\) \(P_{min} = 2 \cdot (0.10 \, \mathrm{M}) \cdot (0.0821 \, \mathrm{L \, atm/mol \, K)} \cdot (293.15 \, \mathrm{K})\)

Calculate the minimum pressure

By calculating the given values, we get the minimum pressure: \(P_{min} = 2 \cdot 0.10 \cdot 0.0821 \cdot 293.15\) \(P_{min} = 4.79 \, \mathrm{atm}\) The minimum pressure that must be applied to purify the water by reverse osmosis is approximately 4.79 atm.

Key concepts

Van't Hoff Factor

The van't Hoff factor \((i)\) is a crucial aspect when dealing with solutions that contain solutes, especially those that dissociate in water. It represents the number of particles a compound separates into in a solution. For non-electrolytes, which do not dissociate, the factor is typically 1. However, with electrolytes like NaCl, it's a different story.
When NaCl dissolves in water, it dissociates completely into sodium \((\mathrm{Na}^+)\) and chloride \((\mathrm{Cl}^-)\) ions. Therefore, two particles emerge from the dissolution of one NaCl molecule. As a result, the van't Hoff factor for NaCl is 2. This factor directly affects calculations of various colligative properties, including osmotic pressure, which is vital for processes like reverse osmosis.
Understanding the van't Hoff factor allows us to correctly account for the concentration and behavior of particles in solutions, ensuring precise calculations in practical applications.

Ideal Gas Constant

The Ideal Gas Constant \((R)\) finds its importance in various gas laws, bridging multiple units and measurements. In the context of reverse osmosis, it is used to relate molarity, temperature, and pressure in the system. The constant has the unit \(0.0821 \text{ L atm/mol K}\), indicating its applicability for calculations involving gases, although it finds use in liquid systems due to its unit versatility.
In solution-related calculations, particularly where pressure and molarity intersect such as in osmosis, \(R\) allows for expressing energy relationships and drives our potential to predict osmotic pressures accurately. It effectively cross-links moles of substance, energy (in terms of pressure and volume), and temperature.
Leveraging \(R\) provides a bridge between theoretical understanding and real-world applications in processes like desalination, reinforcing its crucial role across different areas of chemistry.

Water Desalination

Desalination is a process that removes minerals and salts from seawater or brackish water, making it potable. Reverse osmosis is one of the most efficient methods for desalination. It involves applying pressure to overcome natural osmotic pressure, allowing pure water to pass through a semipermeable membrane while retaining salt and other impurities.
This technique is critical in regions with limited fresh water supplies, offering a viable solution for obtaining drinkable water. The efficiency of this process is significantly influenced by the osmotic pressure needed, which depends on the concentration of dissolved salts. Accurately calculating minimum pressures needed for reverse osmosis is essential, and this often involves concepts like the van't Hoff factor and ideal gas calculations.
Understanding how these elements interact is key to optimizing desalination processes, ensuring sustainability and efficiency in producing clean water across the globe.

Electrolyte Dissociation

Electrolyte dissociation refers to the process where electrolyte compounds, when dissolved in water, split into ions. This splitting is essential because it affects the solution's properties such as conductivity, boiling point, and osmotic pressure.
Taking sodium chloride \((\mathrm{NaCl})\) as an example, it disassociates into sodium \((\mathrm{Na}^+)\) and chloride \((\mathrm{Cl}^-)\) ions in water. Because NaCl dissociates completely, it is termed a "strong electrolyte." This complete dissociation directly influences the calculation of the osmotic pressure in processes like reverse osmosis.
Recognizing whether a substance is a weak or strong electrolyte helps in accurately predicting the behavior of a solution. With complete dissociation, stronger electrolytes like NaCl contribute to a higher van't Hoff factor, impacting the pressure calculations vital for reverse osmosis in desalination plants.