assignment Homework
Active transport Concentration Chemical potential
Thermal and Statistical Physics 2020
The concentration of potassium
\(\text{K}^+\) ions in the internal sap of a plant cell (for example,
a fresh water alga) may exceed by a factor of \(10^4\) the
concentration of \(\text{K}^+\) ions in the pond water in which the
cell is growing. The chemical potential of the \(\text{K}^+\) ions is
higher in the sap because their concentration \(n\) is higher there.
Estimate the difference in chemical potential at \(300\text{K}\) and
show that it is equivalent to a voltage of \(0.24\text{V}\) across the
cell wall. Take \(\mu\) as for an ideal gas. Because the values of the
chemical potential are different, the ions in the cell and in the pond
are not in diffusive equilibrium. The plant cell membrane is highly
impermeable to the passive leakage of ions through it. Important
questions in cell physics include these: How is the high concentration
of ions built up within the cell? How is metabolic energy applied to
energize the active ion transport?
- David adds
-
You might wonder why it is even remotely plausible to consider the
ions in solution as an ideal gas. The key idea here is that the ideal
gas entropy incorporates the entropy due to position dependence, and
thus due to concentration. Since concentration is what differs between
the cell and the pond, the ideal gas entropy describes this pretty
effectively. In contrast to the concentration dependence, the
temperature-dependence of the ideal gas chemical potential will not be
so great.