A vessel of volume contains molecules of an ideal dilute solute dissolved in water

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Physical foundations of the living cell · Physical foundations of the living cell

Life at Thermal Energy, Ch Entropy, free energy and the forces that counting produces

Problem

A vessel of volume

V

contains

N

molecules of an ideal dilute solute dissolved in water, at uniform concentration

c = N/V

. One wall of the vessel is a membrane that passes water freely but holds back the solute completely; the membrane can move so as to change

V

. ((a)) Using the counting argument that a single dilute solute molecule's share of space is

1/c

, derive

\Delta G = k_BT\ln(c_2/c_1)

for moving one such molecule from a region of concentration

c_1

to a region of concentration

c_2

, stating why the enthalpy change is taken to be zero. ((b)) Hence, by writing the free energy of the

N

solute molecules as a function of

V

and considering the membrane moving by a small

\mathrm{d}V

, derive

\Pi = ck_BT

for the osmotic pressure that the solute exerts on the membrane.

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