A) 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 ug of dissolved solids per milliliter. Reverse osmosis units are marketed in this area of soften water. A typical unit exerts a pressure of 8.00 atm and can produce 45.0 liters of water per day. Assuming that all the dissolved solids are MgCO3 and assuming a temperature of 27⁰C, what total volume of water must be processed to produce 45.0 liters of pure water? (Hints: As pure water is produced, the remaining solution becomes more concentrated in the solute and its osmotic pressure increases. Reverse osmosis stops when the solution’s osmotic pressure reaches the applied pressure. The total number of moles of solute initially present in the total volume must remain in the residual solution.)
B) Would the same system described in part A work for purifying seawater? (Assume the seawater is 0.60 M NaCl.) Why or why not?
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To solve part A of the question, we need to determine the total volume of water that must be processed to produce 45.0 liters of pure water.
1. Calculate the total pressure exerted by the dissolved solids in the tap water:
- Using the ideal gas law (PV = nRT), we can rearrange the equation to solve for the number of moles of solute (n).
- Rearranging the equation, we have n = PV / RT.
Given:
- P = 8.00 atm (applied pressure)
- V = volume of water = 45.0 L = 45,000 mL
- R = ideal gas constant = 0.0821 L·atm/(mol·K)
- T = 27°C = 300 K (remember to convert to Kelvin)
Calculate:
n = (8.00 atm) * (45,000 mL) / (0.0821 L·atm/(mol·K) * 300 K)
2. Convert the moles of solute (MgCO3) to grams:
- The molar mass of MgCO3 is: 24.31 g/mol (Mg) + 12.01 g/mol (C) + 3(16.00 g/mol) (O)
- Multiply the number of moles (n) obtained in step 1 by the molar mass to get the mass in grams.
3. Convert grams to milligrams:
- Multiply the mass in grams obtained in step 2 by 1000 to get the mass in milligrams.
4. Convert milligrams of solute to milliliters of total volume:
- Divide the mass in milligrams obtained in step 3 by the concentration of the solution, given as 560 μg (micrograms) of dissolved solids per milliliter.
Now, to solve part B:
Reverse osmosis works by applying pressure to overcome the osmotic pressure of the solute in a solution. In the case of seawater (0.60 M NaCl), the presence of salt significantly increases the osmotic pressure. This means that a higher applied pressure would be required to purify seawater compared to tap water with dissolved solids mostly composed of MgCO3.
Additionally, pure water is more permeable to water molecules than to ions such as Na+ and Cl-. Therefore, even if the applied pressure is sufficient to overcome the osmotic pressure, the reverse osmosis system may not be able to effectively remove all the NaCl ions from seawater.
In summary, the same reverse osmosis system described in part A may not work for purifying seawater due to the higher osmotic pressure caused by the higher concentration of NaCl and the selective permeability of the membrane to ions.