Question

Based upon the technique of reverse osmosis, the approximate minimum pressure required to desalinate sea water containing \(2.5 \%\) (w/v) \(\mathrm{NaCl}\) at \(27^{\circ} \mathrm{C}\) should be (a) \(10.5 \mathrm{~atm}\) (b) \(21 \mathrm{~atm}\) (c) \(2.1 \mathrm{~atm}\) (d) \(1.05\) atm

Short answer

The approximate minimum pressure required to desalinate sea water is \(21 \mathrm{~atm}\).

Step-by-step solution

Understand Reverse Osmosis

Reverse osmosis is a process to remove solutes from solvent in a solution by applying pressure on the side with the dissolved solute. For desalination of seawater, it's necessary to apply a pressure that exceeds the osmotic pressure of the saline water.

Calculate Osmotic Pressure

The osmotic pressure can be calculated using the formula \(\Pi = iCRT\), where \(\Pi\) is the osmotic pressure, \(i\) is the van 't Hoff factor for \(\mathrm{NaCl}\), which is 2 because \(\mathrm{NaCl}\) dissociates into \(\mathrm{Na}^+\) and \(\mathrm{Cl}^-\), \(C\) is the molar concentration of the solution, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Convert Percentage to Molarity

First, convert the temperature from Celsius to Kelvin: \(T = 27 + 273.15 = 300.15 \mathrm{K}\). Then, convert the weight/volume percentage to molarity. Assuming the density of seawater is close to water, \(2.5\% \mathrm{w/v}\) \(\mathrm{NaCl}\) is equivalent to \(25 \mathrm{g}\) of \(\mathrm{NaCl}\) in \(1 \mathrm{L}\) of solution. The molar mass of \(\mathrm{NaCl}\) is approximately \(58.44 \mathrm{g}\)/mol, so \(C = \frac{25}{58.44}\) mol/L.

Plug Values into the Osmotic Pressure Formula

Using the values, the osmotic pressure is \(\Pi = 2\times\frac{25}{58.44}\times0.0821\times300.15\), with \(R = 0.0821 \mathrm{L \cdot atm \cdot K^{-1} \cdot mol^{-1}}\).

Calculate the Pressure

Now calculate \(\Pi\) using the plugged-in values to find the osmotic pressure, which is the minimum pressure required for reverse osmosis to desalinate the seawater.

Key concepts

Osmotic Pressure Calculation

Osmotic pressure is a crucial concept in understanding reverse osmosis desalination. It's the pressure required to prevent the flow of a solvent across a semipermeable membrane separating two solutions of different concentrations. To calculate osmotic pressure, you need to use the formula:
\[\begin{equation}\Pi = iCRT\end{equation}\]
In this formula,

• \(\Pi\) stands for the osmotic pressure,
• \(i\) is the van 't Hoff factor,
• \(C\) is the molarity of the solution,
• \(R\) represents the ideal gas constant, and
• \(T\) is the temperature in Kelvin.

With these variables, you can determine the minimum pressure needed for desalination via reverse osmosis. Understanding each variable's role is critical for accurate calculations.

Van 't Hoff Factor

The van 't Hoff factor, represented by 'i', is key to determining the osmotic pressure in solutions where electrolytes dissolve into ions. For instance, common salt (\(\mathrm{NaCl}\)) dissociates in water to form sodium (\(\mathrm{Na}^+\)) and chloride (\(\mathrm{Cl}^-\)) ions.

The van 't Hoff factor for \(\mathrm{NaCl}\) is 2 because it splits into two ions. However, this factor can vary depending on the degree of dissociation and could be different for other substances. A correct van 't Hoff factor is essential to accurately calculate the osmotic pressure, which influences the force needed to perform reverse osmosis desalination.

Molarity Conversion

Molarity, denoted as 'M', measures the concentration of a solute in a solution. It's defined as moles of solute per liter of solution. Conversion from weight/volume percentage to molarity is a common step in solution-based calculations.

For reverse osmosis desalination scenarios, knowing the molarity of the saline water is necessary to calculate the osmotic pressure. To convert from a percentage to molarity, you need the solute's mass in grams and the solution's volume in liters, as well as the solute's molar mass.

For example, a \(2.5\frac{%}{w/v}\) solution of \(\mathrm{NaCl}\) in water signifies that you have \(2.5\) grams of salt per \(100\) milliliters of water. This information, alongside the molecular weight of \(\mathrm{NaCl}\), allows you to convert the concentration into moles per liter, yielding the molarity that can be used in the osmotic pressure equation.