DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
485 
Evaporation. 
§ 6. The first case we shall take is that of evaporation. Let us suppose that we 
have a liquid and its vapour in a closed vessel, and endeavour to find an expression 
for the density of the vapour when it is in equilibrium with the liquid. We have 
here two systems for which we have to find expressions for L when in a steady state, 
the first being the gas, the second the liquid. 
The variation we shall consider is that which would be produced if a small quantity 
of the liquid were vaporised, keeping the velocities of the molecules the same as in 
the liquid condition, and thus keeping the temperature of the liquid and gas constant. 
We must find the effect of this change on the value of L for the gas and the liquid. 
To do this for the gas, let us consider the case of a cylinder furnished with a piston 
and containing a given quantity of gas. Let x denote the distance of the piston from 
the base of the cylinder, and let us look on the gas as a dynamical system defined by 
the coordinate x. 
We have, by Lagrange’s equations, 
d f?L . p , i. i . 
---- external force tending to increase x ; 
dt dx dx 
or, when there is equilibrium, 
— f — dt — average external force tending to increase x. 
J o dx 
Since x does not enter into the limits of integration, 
X — C lt = dL 
o dx dx' 
The average external force tending to increase x is — _pA, where p is the pressure per 
unit area, and A is the area of the piston. 
Thus 
dL 
dx 
If the gas obeys Boyle’s Law, 
p = It p6, 
where p is the density of the gas and 6 the absolute temperature. II v he the volume 
of the gas, and if its mass be unity, we have 
