1899-1900.] Prof. Kuenen on Proof of Gibbs Phase-rule. 317 
Simple Proof of Gibbs’ Phase-rule. By Professor 
Kuenen. 
(Read January 22, 1900.) 
About a year ago, while writing a Text-book on Heat in which 
the use of higher mathematics had to be avoided, it appeared to' 
me that the phase-rule could be rigorously proved by a process 
which does not involve the deduction of the somewhat difficult 
thermodynamical equations used by Gibbs, Planck and others. 
Quite lately, however, I discovered that proofs somewhat similar 
to mine had been previously given by Nernst and Bancroft, and 
I must therefore not be understood to claim originality in this 
paper. Seeing, however, that modern thermodynamics do not 
yet command in this country the interest they deserve, it will not 
be superfluous to draw the attention of the Society to the subject. 
The phase-rule states that when n mutually independent sub- 
stances are in equilibrium in a system of r phases, the system is 
capable of (n-r + 2) independent variations, or, the number of 
independent variable quantities is (n-r+ 2). 
In determining n we must not count separately those substances 
which in all the phases (either separately or in combination with 
others in the ratio in which they occur in the same phase) may 
be formed out of those that have already been counted, with the 
additional understanding that if we obtain different results for the 
total number by counting in a different order, we are to take the 
smallest of the numbers found. 
A system of ammonium chloride and its products of dissocia- 
tion, ammonia and hydrochloric acid, must therefore be con- 
sidered to contain one substance, if the two substances are present 
in equivalent quantities, two substances, if there is a surplus of 
either of the two gases. Calcium carbonate, on the other hand, 
when dissociating, contains two substances, as neither the carbon 
dioxide in the gas-phase nor the calcium oxide can be formed 
out of the calcium carbonate by itself ; two independent sub- 
stances, say carbon dioxide and calcium oxide, are sufficient, as 
the third substance, carbonate, is formed by the combination of 
