Problems
Physiological Solution for Use in Hospitals
The osmotic pressure in bodily fluids of mammals is 7.7 atm at 36 ◦C. Compute the amount of salt (NaCl) that must be added to 1 litre water to give a solution with the same osmotic pressure.
One kilogram of water and 80 grams of NaCl are mixed, the mixture has a temperature of 80 ◦C. Assume the mixture is ideal.
1. Compute the osmotic pressure of the solution.
2. Compute the entropy of mixing.
3. Compute the minimum work required for separation of salt and water.
Osmotic Equilibrium I
A semipermeable membrane which allows only water to pass, divides two containers. There is 1 litre of water in total, and 10 g of NaCl in each container. One container is kept at a pressure of 10 bar and the other at 20 bar; the temperature of both is 300 K.
1. Show that in equilibrium the pressure difference between the two containers equals the difference in osmotic pressures.
2. Set up the equation needed to determine how water is distributed between the two containers, and determine the water masses in both containers.
3. Will the equilibrium change when the temperature is lowered to 20 ◦C? If so, in which direction does water move, low to high pressure, or high to low?
Osmotic Equilibrium II
A semipermeable membrane which allows only water to pass, divides two containers. In thermodynamic equilibrium container A holds 400 g of water and 20 g salt, container B holds 600 g of water and 20 g of salt. Moreover, container A is kept at a pressure of 20 bar; the temperature of both is 300 K.
Determine the pressure in container B.
Osmotic Equilibrium with Temperature Difference I Two piston-cylinder systems are connected by a semipermeable membrane that allows only water to pass. Both cylinders contain water and NaCl. The left cylinder is pressurized to 15 bar and its temperature is maintained at 320 K . The right cylinder is maintained at a temperature of 325 K.
Determine the pressure that must be exerted on the right cylinder so that in chemical equilibrium the mole fraction of salt in both containers is Xs,L = Xs,R = 0.05.
Hint: Careful, the temperature difference affects the chemical potentials! Osmotic Equilibrium with Temperature Difference II
Two piston-cylinder systems are connected by a semipermeable membrane that allows only water to pass. Both cylinders contain water and NaCl. The left cylinder is pressurized to 10 bar and its temperature is maintained at 300 K. The right cylinder is pressurized to 20 bar and its temperature is maintained at 305 K. The mole fraction of salt in the left container is measured as Xs,L = 0.05.
Determine the mole fraction of salt Xs,R in the right cylinder for the case of chemical equilibrium.
Hint: Careful, the temperature difference affects the chemical potentials! Partial Separation of a Binary Gas Mixture I
Some helium is to be separated from an equimolar mixture of argon and helium. For this, the mixture is pressurized to a pressure pM , and then flows past a membrane, which allows only helium to pass. The helium pressure on the back of the membrane is 4 bar. Determine the pressure pM that is necessary to remove 50% of the helium, in the best case.
Cooling Fluid
A cooling fluid consists of a mixture of water and ethylene glycol (C2H6O2, The glycol mass fraction is cg = 0.3. Assume the mixture is
ideal.
1. Determine the mean molar mass of the mixture.
2. The mole volume of glycol is v¯g = 0.056 m . Compute the specific volume of the mixture, and its mole volume.
3. Determine the entropy of mixing for 1 kg of mixture.
4. What is the minimum work required for complete separation at 0 ◦C, per kilogram of mixture?
5. The above cooling fluid is brought into contact with a semipermeable membrane that allows only water to pass. On the other side of the membrane is a salt (NaCl) solution. In equilibrium at 300 K the cooling fluid is at pres- sure pc = 45 bar and the salt solution is at pressure ps = 5 bar. Determine the mole fraction Xs of salt in the salt solution.
Minimum Work for Reverse Osmosis
We computed the work loss in irreversible mixing as T Smix . This is also the minimum work required for separation.
1. Discuss the above statement.
2. Compute the minimum work for the separation of 1 m3 of salt water at 35 ◦C. The saltwater contains 75 g sodium chloride (NaCl, MNaCl per litre and has a mass density of 1060 kg . Remember that NaCl dissociates into Na+ and Cl− ions. Note: You will get different results if you consider splitting into H2O, Na+, Cl−, than if you consider splitting into H2O and NaCl. The difference is the entropy of mixing be- tween sodium and chlorine.
Reversible Mixing
A fabrication process produces a salty waste flow (density: 1040 g , 50 g of salt per litre). How much work could be obtained by mixing 1 litre of seawater (density: 1025 g flows are at 8 ◦C. , 32 g of salt per litre) with 1 litre of the waste? Assume all
Desalination in Piston-Cylinder Device
2 litres of saltwater are enclosed in a piston cylinder device. The saltwater is compressed up to the pressure p2, which is the pressure at which freshwater just starts to pass a semipermeable membrane. Further increase of pressure forces freshwater through the membrane, until one litre of saltwater remains in the cylinder.
The saltwater contains 50 g sodium chloride (NaCl) per litre and has a mass density of 1040 kg ; it can be considered as incompressible ideal mixture, the temperature remains constant at 15 ◦C.
Assume that the osmotic pressure can be computed from the
1. Compute the pressure p2.
2. Find the relation between the volume remaining in the cylinder and the pressure.
3. Compute the work required for the process.
Reverse Osmosis
Consider the continuous desalination device depicted below. Fresh seawater at 15 ◦C, 1 bar is pumped isothermally to a pressure of 35 bar, and then flows past a semipermeable membrane, which allows only fresh water to pass. The exiting brine drives a turbine, and leaves the system at 1 bar. Assume that the temperature remains constant throughout the device, and that all processes are reversible.
1. Determine the volume flow of freshwater produced per volume flow of sea water.
2. Determine the work required to drive the pump, and the work that can be recovered by the turbine. Compute the net work required per litre of freshwater.
3. Assume now that pump and turbine are irreversible, with efficiencies of 85%, and determine the net work.
Osmotic Power Plant
Saltwater at 15 ◦C, 1 bar is pumped to a pressure of 20 bar and then flows along a semi-permeable membrane through which freshwater enters and dilutes the saltwater. The exiting solution (diluted saltwater) drives a turbine, and leaves the system at 1 bar. Assume that the temperature remains constant throughout the device, and that all processes are reversible.
1. Determine the osmotic pressure of the incoming saltwater.
2. The volume flow of saltwater is 1000 litres . Determine the volume flows of the freshwater drawn in, and of the diluted saltwater that enters the turbine.
Determine the power required to drive the pump, and the power produced by the turbine. Compute the net work produced per litre of freshwater drawn in.