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Chapter 10: Problem 82

A water desalination plant is set up near a salt marsh containing water that is 0.10 \(M\) NaCl. Calculate the minimum pressure that must be applied at \(20 .^{\circ} \mathrm{C}\) to purify the water by reverse osmosis. Assume \(\mathrm{NaCl}\) is completely dissociated.

Short Answer

Expert verified
The minimum pressure that must be applied to purify the water by reverse osmosis at 20°C with a 0.10 M NaCl solution is approximately \(4.79 \, \mathrm{atm}\).

Step by step solution

01

Write down the reverse osmosis equation

The equation for calculating the minimum pressure in reverse osmosis is given by: \(P_{min} = i \cdot c \cdot R \cdot T\) Where: \(P_{min}\) = minimum pressure \(i\) = van't Hoff factor for NaCl (dissociation factor for electrolytes) \(c\) = molar concentration (in moles/liter) \(R\) = Ideal gas constant (0.0821 L atm/mol K) \(T\) = temperature in Kelvin
02

Calculate the van't Hoff factor for NaCl

The van't Hoff factor (i) for NaCl is calculated based on the degree to which it dissociates in solution. Since NaCl completely dissociates into its ions (\(\mathrm{Na}^+\) and \(\mathrm{Cl}^-\)) in the solution, the van't Hoff factor is 2, as there are two ions produced for each salt molecule dissolved.
03

Convert the temperature to Kelvin

To convert the given temperature from Celsius to Kelvin, we add 273.15 to the given value: \(T_{K} = 20^\circ \mathrm{C} + 273.15 = 293.15 \, \mathrm{K}\)
04

Substitute known values into the reverse osmosis equation

Now, we will substitute the known values into the reverse osmosis equation: \(P_{min} = i \cdot c \cdot R \cdot T\) \(P_{min} = 2 \cdot (0.10 \, \mathrm{M}) \cdot (0.0821 \, \mathrm{L \, atm/mol \, K)} \cdot (293.15 \, \mathrm{K})\)
05

Calculate the minimum pressure

By calculating the given values, we get the minimum pressure: \(P_{min} = 2 \cdot 0.10 \cdot 0.0821 \cdot 293.15\) \(P_{min} = 4.79 \, \mathrm{atm}\) The minimum pressure that must be applied to purify the water by reverse osmosis is approximately 4.79 atm.

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Most popular questions from this chapter

a. Use the following data to calculate the enthalpy of hydration for calcium chloride and calcium iodide. $$\begin{array}{|llc|} \hline & \text { Lattice Energy } & \Delta H_{\text {soln }} \\ \hline \mathrm{CaCl}_{2}(s) & -2247 \mathrm{kJ} / \mathrm{mol} & -46 \mathrm{kJ} / \mathrm{mol} \\ \mathrm{Cal}_{2}(s) & -2059 \mathrm{kJ} / \mathrm{mol} & -104 \mathrm{kJ} / \mathrm{mol} \\ \hline \end{array}$$ b. Based on your answers to part a, which ion, \(\mathrm{Cl}^{-}\) or \(\mathrm{I}^{-}\), is more strongly attracted to water?

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The solubility of benzoic acid \(\left(\mathrm{HC}_{7} \mathrm{H}_{5} \mathrm{O}_{2}\right)\) is \(0.34 \mathrm{g} / 100 \mathrm{mL}\) in water at \(25^{\circ} \mathrm{C}\) and is \(10.0 \mathrm{g} / 100 \mathrm{mL}\) in benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) at \(25^{\circ} \mathrm{C}\). Rationalize this solubility behavior. (Hint: Benzoic acid forms a dimer in benzene.) Would benzoic acid be more or less soluble in a 0.1-M NaOH solution than it is in water? Explain.

Reserpine is a natural product isolated from the roots of the shrub Rauwolfia serpentina. It was first synthesized in 1956 by Nobel Prize winner R. B. Woodward. It is used as a tranquilizer and sedative. When 1.00 g reserpine is dissolved in \(25.0 \mathrm{g}\) camphor, the freezing-point depression is \(2.63^{\circ} \mathrm{C}(K_{\mathrm{f}}\) for camphor is \(40 .^{\circ} \mathrm{C} \cdot \mathrm{kg} / \mathrm{mol})\) Calculate the molality of the solution and the molar mass of reserpine.
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