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Fig. 2.13 Thought experiment in the derivation of the osmotic pressure.
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ture and the total volume of the composite system held fixed. Then n a suffers a change dna, and no suffers a change -dna, for, as far as the pure solvent is concerned, the membrane is nonexistent. The volume of the solution changes by the amount va dn o' The work done by the entire composite system is
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dW = P'vo dna (2,47)
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According to the second law this is equal to the negative of the change in free energy dA, given by (2,48) Therefore
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nlRT P'=-novo
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Since n1/n a 1, however, V = n ova is just the volume occupied by the solution. Hence (2,49) It is easy to see that the boiling point of a solution is higher than that of the pure solvent, on account of osmotic pressure. To deduce the change in boiling point, let us first find the difference between the vapor pressure of a solution and that of a pure solvent. This can be done by considering the arrangement shown in Fig. 2.14, which is self-explanatory. Under equilibrium conditions the difference between the pressures is the difference between the pressures of the vapor at B and at C. The pressure at C, however, is the same as that at A, because the vapor is at rest. Hence (2.50) where p' is the mass density of the vapor. On the other hand, the osmotic pressure is by definition equal to the pressure exerted by the column of solution of height h:
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Fig. 2.14 Aid in the derivation of the difference in vapor pressure of a solution and the pure solvent.
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where p is the density of the solution. Dividing (2.50) by (2.51) we have
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which reduces, on using (2.49), to
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where DO is the volume per mole of the solvent. Thus, at a given temperature, the solution has a lower vapor pressure than the pure solvent by the amount (2.53). The meaning of this formula may be made vivid by the qualitative plot of the vapor pressures in Fig. 2.15, from which we immediately see that the solution has a higher boiling point. The rise in boiling point !:!.T can be deduced from the Clapeyron equation, which gives us the slope of the vapor pressure curve of either the solution or the solvent. In the approximation that we are using, these two
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l . - - - - - - - - L - . . L -_ _--+-T T T+t:..T
Fig. 2.15 Difference in boiling point between a solution and the pure solvent.
slopes may be taken to be the same and given by
dP dT
I Tllu
where I and llu both refer to the pure solvent. Therefore
llT= ---'-(dP/dT)
=---I p no Uo
llu p' n 1 RT
We may further make the approximation that the volume per mole of the solvent is negligible compared to that of its vapor, and that the vapor is an ideal gas. RT
RT nlRT no I
We obtain, with these approximations,
where I is the heat of vaporization per mole of the pure solvent.
2.5 THE LIMIT OF THERMODYNAMICS We have seen that a substance in solution exerts osmotic pressure. A solution, in the derivation we have given, is any mixture of substances for which the entropy is greater than the Sum of the entropies of the individual substances before they were mixed. Thermodynamics itself does not tell us what entropy really is, however, and therefore it does not tell us what constitutes a solution and what does not. For example, on purely thermodynamic grounds there is no way to answer the question, "Does a suspension of small particles in water exert osmotic pressure " To answer this question we would have to form a definite opinion concerning the constitution of matter. A nonbeliever in the atomic constitution of matter would hold that a fine suspension of particles does not exert osmotic pressure because it is not qualitatively different from having rocks in water. Even when the rocks are finely pulverized, they are still rocks. Would you think placing a piece of rock in water changes the pressure of the water Atomic theory says that a suspension of fine particle is qualitatively the same as a solution, which is a suspension of atoms. Thus there should be an osmotic pressure.