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 \(\mathrm{L}\) 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 L pure water? b. Would the same system work for purifying seawater? (Assume seawater is 0.60\(M \mathrm{NaCl.} )\)

Short answer

In short, to produce 45 L of pure water, 174.6 L of water must be processed. The same reverse osmosis system would not work for purifying seawater because the osmotic pressure of seawater is much higher than the pressure exerted by the unit.

Step-by-step solution

Calculate the number of moles of MgCO3 in 1L of water

First, we need to find the number of moles of MgCO3 in 1 liter (1000 ml) of water. Moles of MgCO3 = mass of MgCO3 / molar mass of MgCO3 Moles of MgCO3 = (560 * 1000 * 10^{-6} g) / 84.31 g/mol = 6.648 * 10^{-3} moles

Determine the concentration of MgCO3

Now, we can calculate the concentration of the MgCO3 in the water by dividing the number of moles of MgCO3 by the volume of water (in liters): Concentration of MgCO3 = (6.648 * 10^{-3} moles) / 1 L = 6.648 * 10^{-3} mol/L #b. Calculate the volume of water processed#

Calculate the osmotic pressure of MgCO3

We will use the van 't Hoff equation to calculate the osmotic pressure: Osmotic pressure = i * c * R * T where i is the van 't Hoff factor (MgCO3 doesn't dissociate in water, so i = 1), c is the concentration of MgCO3, R is the gas constant (0.0821 L atm/mol K), and T is the temperature in Kelvin (T = 273 + 27 = 300 K). Osmotic pressure = (1) * (6.648 * 10^{-3} mol/L) * (0.0821 L atm/mol K) * (300 K) = 1.6385 atm

Calculate the volume ratio of processed water to pure water

The pressure exerted by the reverse osmosis unit is 8.0 atm. The difference between this pressure and the osmotic pressure is the effective pressure applied for reverse osmosis. Effective pressure = 8.0 atm - 1.6385 atm = 6.3615 atm The volume ratio of processed water to pure water can be calculated using Boyle's law, assuming the dissolved solids have been completely removed: P1 * V1 = P2 * V2 ==> V1 / V2 = P2 / P1 Volume ratio = 6.3615 atm / 1.6385 atm = 3.880

Calculate the volume of water processed

Now we can use this volume ratio to find the volume of water that needs to be processed to produce 45 L of pure water: Volume of processed water = Volume of pure water * Volume ratio Volume of processed water = 45 L * 3.880 = 174.6 L #b. Analyze if the same system works for purifying seawater#

Determine the osmotic pressure of seawater

We will again use the van 't Hoff equation, but now assume that seawater is 0.60 M NaCl: Osmotic pressure = i * c * R * T Since NaCl dissociates into two ions (Na+ and Cl-), i = 2. Osmotic pressure = (2) * (0.60 mol/L) * (0.0821 L atm/mol K) * (300 K) = 29.587 atm

Compare osmotic pressures and analyze if the system would work

The osmotic pressure of seawater is much higher than the osmotic pressure of tap water (29.587 atm vs. 1.6385 atm). Thus, the reverse osmosis unit with a pressure of 8.0 atm wouldn't be effective for purifying seawater since the applied pressure is much lower than the osmotic pressure of seawater. In conclusion, it is necessary to process a total volume of 174.6 L of water in order to produce 45 L of pure water. The same system wouldn't be effective for purifying seawater as its osmotic pressure is much higher than the pressure exerted by the reverse osmosis unit.

Key concepts

Osmotic Pressure

Osmotic pressure is a key concept in understanding how solutions and solvents interact through a semi-permeable membrane. Imagine the membrane as a gatekeeper, letting some molecules pass while keeping others out. Osmotic pressure arises when there is a difference in solute concentration on either side of this membrane. It is the pressure required to stop the solvent (like water) from passing through the membrane to balance the concentrations.

Factors influencing osmotic pressure include:

• The type of solute and its concentration
• The temperature of the solution
• The dissociation of the solute into ions (if applicable)

A higher solute concentration means higher osmotic pressure, as more pressure is needed to prevent the flow of the solvent through the membrane. This principle is crucial in reverse osmosis systems, which use pressure to purify water by overcoming the osmotic pressure of dissolved impurities.

Water Purification

Water purification involves removing impurities and contaminants to make the water suitable for various uses, especially drinking. Reverse osmosis is a popular method of water purification. It works by applying pressure to push water through a semi-permeable membrane, leaving unwanted particles behind.

Here are some benefits of using reverse osmosis for purification:

• Removes a wide range of contaminants
• Produces clean and safe drinking water
• Reduces the hardness of water

Reverse osmosis is particularly beneficial in areas with hard water, containing high levels of dissolved minerals like calcium and magnesium. By forcing water through a membrane, such systems not only soften the water but also ensure its overall quality by removing other dissolved solids and impurities effectively.

Molarity

Molarity is a measure of the concentration of a solute in a solution, expressed as moles of solute per liter of solution. It’s a fundamental concept in chemistry that helps us understand the strength or concentration of solutions.

The formula to calculate molarity is:\[ M = \frac{n}{V} \]Where:

• \( M \) is the molarity
• \( n \) is the number of moles of solute
• \( V \) is the volume of solution in liters

Understanding molarity is crucial in calculating osmotic pressure using the Van 't Hoff equation because it dictates how strong or weak the solvent interaction is based on solute concentration. In water purification processes, knowing the molarity of dissolved solids helps in designing systems that effectively use reverse osmosis to separate and purify water efficiently.

Van 't Hoff Equation

The Van 't Hoff equation is essential in calculating the osmotic pressure of solutions. It provides the formula needed to determine how pressure changes with solute concentration in a solution. The equation is:\[ \Pi = iCRT \]Where:

• \( \Pi \) is the osmotic pressure
• \( i \) is the van 't Hoff factor, which accounts for ion dissociation
• \( C \) is the molarity of the solute
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

In reverse osmosis systems, understanding how to apply the Van 't Hoff equation can determine if the applied pressure is adequate to purify water. For example, purifying seawater is challenging because its high salt concentration results in high osmotic pressure. The Van 't Hoff equation helps in designing effective systems to generate sufficient pressure to counteract this, ensuring efficient purification.