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Chapter 13: Problem 107

A very long pipe is capped at one end with a semipermeable membrane. How deep (in meters) must the pipe be immersed into the sea for fresh water to begin to pass through the membrane? Assume the water to be at \(20^{\circ} \mathrm{C}\) and treat it as a \(0.70 \mathrm{M} \mathrm{NaCl}\) solution. The density of seawater is \(1.03 \mathrm{~g} / \mathrm{cm}^{3}\) and the acceleration due to gravity is \(9.81 \mathrm{~m} / \mathrm{s}^{2}\)

Short Answer

Expert verified
Convert the osmotic pressure from atm to Pascals (Pa) by multiplying it by 101325, since 1 atm = 101325 Pa. Use these values for \(P\), \(\rho\), and \(g\) in the formula \(h=P/(\rho g)\) to get the depth of the pipe in meters.

Step by step solution

01

Calculate the osmotic pressure of the sea water

First, calculate the osmotic pressure of the sea water using the formula of osmotic pressure \(P=iMRT\). The van 't Hoff factor (\(i\)) for a NaCl solution is 2. The molarity (\(M\)) of the sea water is given as 0.70 M. The universal gas constant (\(R\)) is 0.0821 L atm/mol K. The temperature (\(T\)) must be converted from Celsius to Kelvin by adding 273 to the given temperature in Celsius. Therefore, the osmotic pressure of the sea water is \(P=(2)(0.70 M)(0.0821 L atm/mol K)(20^{\circ}C + 273)\).
02

Equate the osmotic pressure with the pressure inside the pipe

Next, equate the osmotic pressure with the pressure at the bottom of the pipe. The pressure inside the pipe can be calculated as \(P=\rho gh\). The density of sea water (\(\rho\)) is given as 1.03 g/cm\(^3\), but it should be converted to kg/m\(^3\) by multiplying it by 1000. The acceleration due to gravity (\(g\)) is given as 9.81 m/s\(^2\). Let the depth of the pipe (\(h\)) be the unknown that we are solving for. The equation then becomes \(h=P/(\rho g)\).
03

Solve for the depth of the pipe

Finally, solve for the depth of the pipe (\(h\)). The value of \(h\) gives the minimum depth that the pipe must be immersed into the sea for fresh water to begin to pass through the membrane.

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