Where does it all end?
So, a quick recap. We can figure out which way water is flowing. And, using the iteration technique at the end of the section on diffusion through a membrane, we can figure out how much water is flowing at each time step (delta t). You may remember that we stopped figuring it out after 5 or 6 time steps because the rate of flow kept slowing down, so although we were getting to equilibrium, we were getting there slower and slower and s-l-o-w-e-r. It got boring.
However, there is still one more property of the system that we can figure out -- what happens at the end of that slower and slower process -- in other words, what are the equilibrium concentrations, and how much water is on either side of the membrane. And we can do this without endlessly pushing buttons on a calculator.
Let's use sugar as an example. Say we started with 0.2 moles of sugar in 1 liter of water on one side, and 0.8 moles of sugar in 1 liter of water on the other side. The water is free to move back and forth across the membrane, but the sugar's not going anywhere, and no extra sugar or water can move in or out of the 2-compartment system. So at the end of the day,
•there will still be 0.2 moles of sugar on one side and 0.8 on the other side,
•and there will still be 2 liters of water,
•and the concentrations on either side will be equal,
•BUT the water will not be equally distributed.
How can you find the final water levels using intuition?
- I need a hint ... : The final water level on the 0.8 mole side must be
4 times higher than on the 0.2 mole side.
- ...another hint ... : The total water level has to add up to 2 liters
I think I have the answer: 1.6 L+ 0.4 L fulfills these conditions
How can you find the final water levels using formal equations?
- I need a hint ... : the final molarities must be equal, so
0.2/W1 = 0.8/W2 , or
0.2 W2 = 0.8 W1 ,or
W2 = 4 W1
- ...another hint ... : The total amount of water is 2 L so W1
+ W2 = 2
I think I have the answer: We can substitute 4 W1
for W2, so W1 + 4 W1 = 2
5 W1 = 2
W1 = 0.4 L
W2 = 2 - 0.4 = 1.6 L
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