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 atm and can produce 45 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 L pure water? b. Would the same system work for purifying seawater? (Assume seawater is 0.60 $M$ NaCl.)

Short answer

To produce 45 L of pure water, the system must process a total of 91.4 L of tap water containing dissolved ${\mathrm{MgCO}}_3$. The same system would not work for purifying seawater, as the osmotic pressure required (14.8 atm) is greater than the pressure applied by the reverse osmosis unit (8.0 atm).

Step-by-step solution

Part (a): Calculate the total dissolved solids in 45 L

First, we need to find out how many grams of ${\mathrm{MgCO}}_3$ are present in 45 L of water. Since the tap water contains 560 μg of dissolved solids per milliliter, we can use this information to calculate the total dissolved ${\mathrm{MgCO}}_3$. Total dissolved solids in 45 L of water = 560 μg/mL $\times$ 1000 mL/L $\times$ 45 L = $25.2\times {10}^3$ μg = 25.2 g

Part (a): Calculate the number of moles of ${\mathrm{MgCO}}_3$

Next, we need to find out the number of moles of ${\mathrm{MgCO}}_3$ present in 25.2 g. For this, we will use the molar mass of ${\mathrm{MgCO}}_3$, which is approximately 84.31 g/mol. Number of moles of ${\mathrm{MgCO}}_3$ = ${\displaystyle \frac{25.2\text{g}}{84.31\text{g/mol}}}=0.299\text{mol}$

Part (a): Calculate the osmotic pressure

Now we'll calculate the osmotic pressure, which is the pressure required to stop the net flow of water due to a concentration gradient. For this, we'll use the formula: Osmotic pressure (Π) = nRT/V Here, n is the number of moles of solute, R is the ideal gas constant ($0.0821\mathrm{L}\mathrm{atm}{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$), T is the temperature in Kelvin (273 + 27 = 300 K), and V is the volume of water in liters. Osmotic pressure (Π) = ${\displaystyle \frac{(0.299\mathrm{mol})(0.0821\mathrm{L}\mathrm{atm}{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1})(300\mathrm{K})}{V}}$ Using the information provided in the problem, the reverse osmosis unit exerts a pressure of 8.0 atm. So, we'll set the osmotic pressure equal to 8.0 atm and solve for V: 8.0 atm = ${\displaystyle \frac{(0.299\mathrm{mol})(0.0821\mathrm{L}\mathrm{atm}{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1})(300\mathrm{K})}{V}}$

Part (a): Calculate the total volume of water

To find the total volume of water processed, we'll solve for V in the above equation: V = ${\displaystyle \frac{(0.299\mathrm{mol})(0.0821\mathrm{L}\mathrm{atm}{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1})(300\mathrm{K})}{8.0\mathrm{atm}}}=91.4\mathrm{L}$ So, the system must process a total of 91.4 L of water to produce 45 L of pure water.

Part (b): Check if the system works for purifying seawater

To check if the system works for purifying seawater (assuming 0.60 M NaCl), we need to calculate the osmotic pressure required to remove NaCl. Osmotic pressure (Π) = MRT Here, M is the molarity of the solute (0.60 M), R is the ideal gas constant (0.0821 L atm K⁻¹ mol⁻¹), and T is the temperature in Kelvin (300 K). Osmotic pressure (Π) = $(0.60\mathrm{M})(0.0821\mathrm{L}\mathrm{atm}{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1})(300\mathrm{K})=14.8\mathrm{atm}$ Since the osmotic pressure required for purifying seawater (14.8 atm) is greater than the pressure applied by the reverse osmosis unit (8.0 atm), the system would not work for purifying seawater.

Key concepts

Osmotic Pressure

Imagine you have two containers separated by a semi-permeable membrane. One contains pure water and the other contains water mixed with some dissolved substance, like salt. Water naturally moves from the pure side to the side with the dissolved substance to balance the concentrations. This movement exerts a pressure known as osmotic pressure. Osmotic pressure is crucial when we talk about reverse osmosis, where we use pressure to move water in the opposite direction, from the side with the solute to the pure side. This process is used to make drinking water from sources such as hard water or even seawater. In the exercise, osmotic pressure is calculated using the formula: $$\mathrm{\Pi }=\frac{nRT}{V}$$where:

• $\mathrm{\Pi }$ is the osmotic pressure.
• $n$ is the number of moles of solute.
• $R$ is the ideal gas constant (0.0821 L atm K⁻¹ mol⁻¹).
• $T$ is the temperature in Kelvin.
• $V$ is the volume in liters.

By knowing these variables, we can determine the pressure needed to filter water, like in reverse osmosis systems.

Moles Calculation

Moles in chemistry are like a counting unit. Just as a dozen equals 12 items, a mole equals approximately $6.022\times {10}^{23}$ chemical entities or particles. This is a huge number but it allows chemists to count atoms and molecules in macroscopic quantities.In the context of our exercise, we needed to calculate how many moles of ${\mathrm{MgCO}}_3$ were in the water to figure out the osmotic pressure. We start with the mass of the dissolved solids, which in this case was 25.2 g of ${\mathrm{MgCO}}_3$. Using the molar mass of ${\mathrm{MgCO}}_3$ (approximately 84.31 g/mol), we calculate the moles using:$$\text{Number of moles}=\frac{\text{mass in grams}}{\text{molar mass in g/mol}}$$So for ${\mathrm{MgCO}}_3$, it would be:$$\text{Number of moles}=\frac{25.2\text{g}}{84.31\text{g/mol}}\approx 0.299\text{mol}$$Knowing the number of moles is crucial because it allows us to use the osmotic pressure formula to determine if the system can effectively purify the water.

Hard Water

Hard water contains high levels of dissolved minerals, primarily calcium and magnesium compounds. Areas with hard water, like the Southwest United States, face challenges with scaling and appliances wearing out faster. This is because minerals in the hard water accumulate in pipes and appliances. Reverse osmosis is a widely used method to "soften" hard water, by filtering out these dissolved solids. By applying pressure to force water through a membrane, impurities like magnesium carbonate are removed, leaving behind purified water. Although the exercise considered only magnesium carbonate as the dissolved solid, real hard water can contain a variety of minerals and impurities. It's important to note that reverse osmosis systems are energy-dependent and can vary in their effectiveness based on the mineral content of the water. While reverse osmosis works well for hard water, it requires greater pressure to purify saltier solutions, such as seawater.