Why does the cell swell when the Na/K pump is inhibited?

Question

Looking for a graduate/PhD level explanation

Answer

We have to learn about the electrochemical gradient, and then have a look at two states: a cell without ATP pumps, and a cell with.  The purpose of the ATP pump is to maintain the electrochemical gradient, and is estimated to account for 60-70% of our daily energy expenditure (~1400 calories).  Therefore, the cell without the pumps will not have an electrochemical gradient.  We first need to appreciate what the electrochemical gradient is and why it needs to be maintained.
Lets look at the physics of the electrochemical gradient.  There is a -70mV potential difference inside the cell (ICF), which means it has less positive charge than the extracellular fluid (ECF).  This potential means there is a desire for either:

positive charge to flood into the cell
negative charge to run out of the cell

This potential is no different than batteries (e.g., galvanic cells) or capacitors, and it wants to go to zero, to have no difference in potential.  So whats keeping it like that?
The ATP pump is the obvious and most important answer (causing Na+ dominating in ECF; K+ dominating in ICF), but there are other factors.  The negative charge outside (ECF) is made up of chloride (Cl-), with anionic proteins making a sizable but significantly smaller contribution.  The proteins are incapable of crossing the cell membrane, whereas chloride can.  The ICF has a variety of anions, not so much from proteins, though.  The key is that while there is Cl-, concentration-wise, its out of sync with the outside.
So, the "electro-" part in electrochemical is obvious: the number of cations and anions is out of sync.  What about the "-chemical" part?  Well, thats what the above paragraph was getting at.  The interactions between water, its solvents, and the solvents between one another generates a complex interaction that is still being studied and figured out today.  To keep it short, though, while electrically, Cl- and PO4- both are just a negative charge and generate a single negative charge, their other differences (e.g., structure, strain, bond energy, dipole moment frequency, etc.) are responsible for the chemical forces generated.
When you step back, however, you can look at the chemical interactions as just another force that wants to go to "zero difference" just like the electrical one.  For further clarification, the chemical forces are exactly the same as what makes concentration cells work.  Anyway, these two forces are at play.  Two examples: the chemical force of Cl- and proteins want to go into the cell; the large electrical force is compelling Na+ into the cell as well.
Okay, now we can go back to our ATP pump-less cell.  Remember, the cell membrane is "leaky," and water, small ions, and large lipophilic molecules can cross, but large, charged proteins cannot.  So, the most obvious candidate without a pump is Na+:  it is being pushed to the negative charge, and also along its chemical gradient (ECF -> ICF).  K+ only slowly escapes the cell because its concentration gradient is at odds with the electrical desire to stay inside the negative cell.  As Na+ enters, however, K+ improves its ability to leave since the electrical pull is reduced.
You get the idea, and you can play out in your head each component vying to reach electrochemical equilibrium.  Now before we get to the swelling part, which is actually simple (but you wanted a graduate level answer!), lets look where this mess "ends."  The end point is not zero chemical and electrical forces, but a "net" force of zero, where a small electrical force counters a equal but opposite force.  Remember the negatively-charged ECF proteins?  They can never satisfy their chemical gradient (into the cell).  Their negative charge pushes some anions against their gradient, causing a negative charge inside the cell.  So, the cell, thanks to the membrane, will always be slightly negative inside.
Now, the "water follows sodium" rule is not a rule and more a consequence of the ATP pump.  Water also follows its concentration gradient, and sodium just happens to be the most abundant solvent.  So this abundant Na+ floods into the cell when the ATP pump is inactive, causing water to follow the increased concentration inside the cell.  With the ATP pump on, the greater concentration of solutes was in the ECF.  Without it, the ECF and ICF will have the same concentration, but compared to a cells normal state, that means theres a higher concentration of solutes than there used to be, causing a swelling because water wants to have equal concentrations (aka osmolarities) in all communicating compartments.

Answer

The sodium potasium pump helps to control cell volume. If this one fails the osmolarity outside the cell is higher and then water moves by osmosis through the cell membrane in order to restablish homeostasis

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Why does the cell swell when the Na/K pump is inhibited?

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