Page:Scientific Papers of Josiah Willard Gibbs.djvu/140

It may easily be shown that the same must be true in the limiting cases in which $T=0$ and $P=0$. For negative values of $P$, (133) is always capable of negative values, as its value for a vacuum is $Pv$.

For any body of the temperature $T$ and pressure $P$, the expression (133) may by (91) be reduced to the form

$\zeta -M_{1}m_{1}-M_{2}m_{2}...-M_{n}m_{n}.$

(135)

We have already seen (page 77) that an expression like (133), when $T,P,M_{1},M_{2},...M_{n}$ and $v$ have any given finite values, cannot have an infinite negative value as applied to any real body. Hence, in determining whether (133) is capable of a negative value for any phase of the components $S_{1},S_{2},...S_{n}$, and if not, whether it is capable of the value zero for any other phase than that of which the stability is in question, we have only to consider the least value of which it is capable for a constant value of $v$. Any body giving this value must satisfy the condition that for constant volume

$d\epsilon -Td\eta -M_{1}dm_{1}-M_{2}dm_{2}...-M_{n}\geqq 0,$

(136)

or, if we substitute the value of de taken from equation (86), using subscript $a,...g$ for the quantities relating to the actual components of the body, and subscript $h,...k$ for those relating to the possible,

$td\eta +\mu _{a}dm_{a}...+\mu _{g}dm_{g}+\mu _{h}dm_{h}...+\mu _{k}dm_{k}-Td\eta -M_{1}dm_{1}-M_{2}dm_{2}...-M_{n}dm_{n}\geqq 0.$

(137)

That is, the temperature of the body must be equal to $T$, and the potentials of its components must satisfy the same conditions as if it were in contact and in equilibrium with a body having potentials $M_{1},M_{2},...M_{n}$. Therefore the same relations must subsist between $\mu _{1},\mu _{2},...\mu _{n}$ and $M_{1},M_{2},...M_{n}$ as between the units of the corresponding substances, so that

$\mu _{a}m_{a}..+\mu _{g}m_{g}=m_{1}M_{1}..+m_{n}M_{n},$

(138)

and as we have by (93)

$\epsilon =t\eta -pv+\mu _{a}m_{a}..+\mu _{g}m_{g},$

(139)

the expression (133) will reduce (for the body or bodies for which it has the least value per unit of volume) to

$(P-p)v,$

(140)

the value of which will be positive, null, or negative, according as the value of

$P-p$

(141)

is positive, null, or negative.

Hence, the conditions in regard to the stability of a fluid of which all the ultimate components are independently variable admit a very simple expression. If the pressure of the fluid is greater than that