54 



NA TURE 



[November i8, 1897 



As is well known, this theorem was originally proved by 

 Van 't Hoff by employing a differential thermodynamical pro- 

 cess, which led to the result vdp — V(iP. Assuming Henry's 



law in the form — = const., and Boyle's law for both gas and 



solution, i.e. pv = const., and PV = const., the above result 

 follows at once. Substantially the same proof was given by 

 Nernst in his " Theoretische Chemie." 



Quite recently, however, a new and novel proof of the same 

 theorem was published by Lord Rayleigh in the columns of 

 Nature. In this proof Lord Rayleigh avoids the assumption 

 of the equation PV = const., and herein lies a defimte advance 

 in the subject. The proof is based on the validity of Boyle's 

 law for the gas, and Henry's law ; but as the solvent is assumed 

 to be involatile, it was objected by Lord Kelvin that the great 

 majority of cases would thereby be excluded. So far as I can 

 see, a small addition to Lord Rayleigh's proof will suffice to 

 free it from this objection. 



Besides this, I think that Lord Rayleigh's proof may be 

 generalised so that even the assumption of Boyle's law for the 

 gas is not required, at all events formally. 



The primary assumption to be made is that for isothermal 

 equilibrium the ratio of the concentrations of the substance in 

 question, as gas and as solute, remains constant. This is usually 

 known as the Distribution-Law, and cannot be regarded as a 

 mere deduction from Boyle's law, and a certain form of state- 

 ment of Henry's law. Recent research rather goes to show 

 that it is a fundamental law of great generality. Accordingly 

 I venture to employ Lord Rayleigh's method of proof, as 

 follows. 



ad and ef are two pistons, e/ being impermeable, and ad 

 permeable for the solvent alone. i>c and d'c' are two fixed walls, 

 of which d'c' is impermeable and dc permeable for the solute 





m)}m}m 



~5 01 L» ri o»x 



V 



Sol 



ren.tl 



^ Jtn/iey>n€,a,k1^ 





^ A)e-r-»»tAAb7« /« S<^v«MC a.7»M<aL. 



only. The piston ad is for the present fixed, and encloses a 

 volume V of solvent between itself and l/c. Suppose the cylinder 

 to have unit section, and denote height of upper piston above 

 the fixed semi-permeable wall by x. The whole process is con- 

 ducted at the constant temperature /. Suppose now that be- 

 tween ef and d'c' there is enclosed a quantity of the solute as 

 gas, of volume v, temperature ( and pressure /, the amount 

 being so chosen that it is just sufficient to saturate the volume 

 V of solvent at this temperature and pressure. Let p denote 

 density of the gas and suppose p = p^(p' (p) to be the isothermal 

 equation of state for the gas, where <(> is an undetermined func- 

 tion. Take as unit of mass the mass of the enclosed gas. Allow 

 the upper piston to rise reversibly to a height x, which is a very 

 great multiple of the initial height. The work done in this 

 process is : — 



/> = *(i) - *(i). 



Let d'c' be removed and the gas reversibly compressed, where- 

 by it is reversibly absorbed, the small amount of irreversibility 

 at the beginning becoming vanishingly small in the limit. De- 



NO. 1464. VOL. 57] 



noting by c the concentration of the solution at any moment, we 

 have during the downward stroke : — 



* px + cV = I 



- = K (Distribution-Law). 

 P 



The work done on the system in this stroke, whereby the gas 

 is just completely absorbed, is given by : — 



/ /'^^ = \<f> {p)T 



But 



-irvi; = -, since by hypothesis -and - are the concentra- 

 KV V ^ ^1 Y V 



tions of the substance in solution and as gas respectively, for 

 equilibrium at t and /. Thus the work done so far by the 

 system is : — 



<7^y) - Hi) 



where x is indefinitely great. 



Separate gas and solvent now, working both pistons so as to 

 keep the concentrations constant, and thus arrive at the initial 

 state, whereby in this portion of the process the system does 

 work pv - PV. 



Since the net work obtainable in a reversible isothermal cycle 

 is zero, we have finally : — • 



Now the term in brackets is zero if (p{z) has the form log z or 

 any positive power of 2, so that it vanishes if <j)(z) has the form 

 a log 3 + dQ + i>-^z + d.2Z'+ &c. Hence it vanishes if (p'(z)ha.s 



the form ~ + bf^ + biZ + b,^z- + &c., since the latter series is by 



hypothesis convergent. 



That is to say, pv = PV if the isothermal equation of 

 state is — 



/=p2^? + 3o + b^p + 3,p2 + &c. y 



or p = ap + bop^ + bip^ + b2p'^+ . . . 



This includes the equations of Boyle and Van der Waals as 

 special cases. 



The equation pv = 'P'V is thus a formal consequence of the 

 distribution-law and the expressibility of / as an infinite power 

 series of p. However, when Boyle's law does not hold, this 

 result loses much of its significance, as it does not then lead to 

 an equation of state for the solution. So that this slight extension 

 of Lord Rayleigh's result is not perhaps of much practical use. 



Holywood, Co. Down. F. G. DONNAN. 



The Law of Divisibility. 



With respect to Mr. Burgess's letter in your issue of Novem^ 

 ber 4, perhaps the following general rule for testing the divisi- 

 bility of a given number by another, which I found some days 

 ago, may be of some interest. 



Any number 



Z = an . io« + «„_iio«-i +... + a^io" +...+ai . 10 -f- ^o 

 is divisible by another number N when the sum 



2(av-a+i . 10— 1 -t-... -f av)(io<' - N)" 



can be divided by N without residue ; otherwise the residue of 

 this division is equal to the residue of the division Z : N. 



Ofcourse, from(io" - N)" the nearest multiple of N must be 

 subtracted. 



Examples : (i) N =7. Take a = i ; then lo" - N = 3, and 

 Z is a multiple of 7 when ^q + 3^1 +2a.2 - a^ - 3«4 - 2^5 -1-... 

 is divisible by 7. 



