  We consider the first-order phase transition of a charged anti-de Sitter
black hole, and find that the equation of state with the conditions of the two
coexisting phases, leads to the two coupled equations about the thermodynamic
volumes of small black hole and large black hole. By solving the equations, it
is found that each reduced volume is only a function of the parameter $\omega$
. All properties of the coexistence curve can be studied from the two volume
functions. In particular, each thermodynamic quantity is described by a
piecewise analytic function. The demarcation point is located at
$\omega_{d}=12(2\sqrt{3}-3)$. The thermodynamic function but not its
derivative, is continuous at the point. This property is completely different
from that of the ven der Waals fluid. Moreover, the thermodynamic behaviors as
$\omega\rightarrow0$ are discussed. From which one can easily obtain some
critical exponents and amplitudes for small-large black hole phase transitions.
