Question

In some regions of the southwest United States, the water is very hard. For example, in Las Cruces, New Mexico, the tap water contains about \(560 \mu \mathrm{g}\) of dissolved solids per milliliter. Reverse osmosis units are marketed in this area to soften water. A typical unit exerts a pressure of \(8.0 \mathrm{~atm}\) and can produce \(45 \mathrm{~L}\) water per day. a. Assuming all of the dissolved solids are \(\mathrm{MgCO}_{3}\) and assuming a temperature of \(27^{\circ} \mathrm{C}\), what total volume of water must be processed to produce \(45 \mathrm{~L}\) pure water? b. Would the same system work for purifying seawater? (Assume seawater is \(0.60 \mathrm{M} \mathrm{NaCl}\).)

Short answer

To produce 45 L of pure water from the hard tap water in Las Cruces, New Mexico, a total volume of 65.5 L of tap water must be processed. The same system would not work for purifying seawater, since the pressure exerted by the unit (8.0 atm) is insufficient to overcome the osmotic pressure of seawater (29.6 atm).

Step-by-step solution

Convert the concentration of dissolved solids into molarity

First, we need to find the molarity of the dissolved solids (MgCO3) in the tap water. For that, we will use the given concentration of solids (560 µg per mL) and the molar mass of MgCO3 (84.31 g/mol). The concentration of MgCO3 in tap water (in moles per L) is: \(\frac{560 \times 10^{-6}~\text{g}}{84.31~\text{g/mol}} \times 10^3 ~\text{mL/L} = 0.00664~\text{M}\)

Calculate the osmotic pressure of the tap water

Now, we will use the vant Hoff equation to calculate the osmotic pressure of tap water: \(\Pi = i \times c \times R \times T\) Where \(Π\) is the osmotic pressure, \(i\) is the vant Hoff factor (i = 2 for MgCO3), \(c\) is the molar concentration, \(R\) is the gas constant (0.0821 L atm/mol K), and \(T\) is the temperature in kelvin (27°C = 300K). \(\Pi = 2 \times 0.00664~\text{M} \times 0.0821~\text{L atm/mol K} \times 300~\text{K} = 3.26~\text{atm}\)

Calculate the total volume of water needed to produce 45 L of pure water

The reverse osmosis unit exerts a pressure of 8.0 atm, so the effective pressure difference it can make is: \(P_\text{diff} = 8.0~\text{atm} - 3.26~\text{atm} = 4.74~\text{atm}\) We know that the unit can produce 45 L of pure water per day, so, to calculate the total volume of water needed, we will use the proportion: \(\frac{\Pi_\text{tap}}{P_\text{diff}} = \frac{45}{V_\text{needed}}\) \(V_\text{needed} = \frac{P_\text{diff}}{\Pi_\text{tap}} \times 45\) \(V_\text{needed} = \frac{4.74~\text{atm}}{3.26~\text{atm}} \times 45~\text{L} = 65.5~\text{L}\) Therefore, to produce 45 L of pure water, a total volume of 65.5 L of tap water must be processed.

Calculate the osmotic pressure of the seawater

Now let's find out if the same system would work for purifying seawater. We will use the given concentration of NaCl in seawater (0.60 M) and the fact that the vant Hoff factor for NaCl is 2. The temperature, gas constant, and the unit pressure remain the same. \(\Pi_\text{seawater} = 2 \times 0.60~\text{M} \times 0.0821~\text{L atm/mol K} \times 300~\text{K} = 29.6~\text{atm}\)

Compare the osmotic pressure with the pressure exerted by the unit

The osmotic pressure of seawater (29.6 atm) is much higher than the pressure exerted by the reverse osmosis unit we have considered (8.0 atm). To purify seawater with this system is not possible since the pressure exerted by the unit would not be enough to overcome the osmotic pressure of seawater. So, this system would not work for purifying seawater.

Key concepts

Osmotic Pressure Calculation

Understanding how reverse osmosis works requires a grasp of osmotic pressure calculation. Osmotic pressure is the force that a solvent, like water, exerts when it moves across a semipermeable membrane to equalize solute concentrations on both sides. This is pivotal in processes like water purification, where reverse osmosis machines apply external pressure to oppose natural osmotic pressure and thus purify water by pushing it through a membrane.

Osmotic pressure, \(\Pi\), can be calculated using the formula \(\Pi = i \times c \times R \times T\), where \(i\) is the van't Hoff factor, \(c\) is the molar concentration of the solute, \(R\) is the ideal gas constant, and \(T\) is the absolute temperature in Kelvin. This equation, stemming from the ideal gas law, implies that the process depends on the number of particles in solution (reflected in \(i\) and \(c\)) and the conditions (reflected in \(R\) and \(T\)). By calculating the osmotic pressure of a solution, we can determine the necessary pressure a reverse osmosis system must apply to effectively purify water.

Van't Hoff Factor

The van't Hoff factor, denoted by \(i\), is a dimensionless number representing the number of particles a compound dissociates into when it’s dissolved in water. For example, \(\mathrm{MgCO}_3\) dissociates into two particles (\(\mathrm{Mg}^{2+}\) and \(\mathrm{CO}_3^{2-}\)), so \(i\) for \(\mathrm{MgCO}_3\) is 2. This information is crucial when using the osmotic pressure formula because the factor affects the total number of solute particles contributing to the osmotic pressure. If \(i\) is not accurately assessed or factored into calculations, the estimated osmotic pressure could be significantly off, leading to a miscalculation of the force needed to purify water through reverse osmosis.

In educational contexts, emphasizing the importance of the van't Hoff factor in osmotic pressure calculations is vital—it underlines the particle nature of solutes and the significance of understanding chemical formulas and their dissociation behavior.

Molarity of Solutions

Molarity is a measure of the concentration of solute in a solution, expressed in moles of solute per liter of solution (mol/L), and is commonly used in chemical calculations, including the operation of reverse osmosis systems. The ability to convert from mass units (like micrograms per milliliter) to molarity is an essential skill in chemistry.

To find the molarity (\(c\)), you divide the mass of the solute by its molar mass and then divide by the volume of the solution in liters. In our exercise, molarity was calculated from the given mass of \(\mathrm{MgCO}_3\) and the volume of water. \(c = \frac{mass \times 10^{-6}}{molar \ mass} \times 10^3\)

When discussing molarity within the concept of reverse osmosis, it becomes clear that the concentration of dissolved substances directly impacts the efficiency of water purification. A higher molar concentration means a higher osmotic pressure that the system must overcome. Therefore, understanding molarity is not just about mastering a computation method; it's also about appreciating its relevance in practical applications such as water treatment.