14
1 The Behavior of Gases and Liquids
(the mass of the system) are extensive variables, whereas P and T are intensive
variables. The quotient of two extensive variables is an intensive variable. The density ρ is defined as m/V , and the molar volume V m is defined to equal V /n. These are
intensive variables. One test to determine whether a variable is extensive or intensive
is to imagine combining two identical systems, keeping P and T fixed. Any variable
that has twice the value for the combined system as for one of the original systems
is extensive, and any variable that has the same value is intensive. In later chapters
we will define a number of extensive thermodynamic variables, such as the internal
energy U, the enthalpy H, the entropy S, and the Gibbs energy G.
We are sometimes faced with systems that are not at equilibrium, and the description
of their states is more complicated. However, there are some nonequilibrium states that
we can treat as though they were equilibrium states. For example, if liquid water at
atmospheric pressure is carefully cooled below 0 ◦ C in a smooth container it can remain
in the liquid form for a relatively long time. The water is said to be in a metastable
state. At ordinary pressures, carbon in the form of diamond is in a metastable state,
because it spontaneously tends to convert to graphite (although very slowly).
Differential Calculus and State Variables
Because a dependent variable depends on one or more independent variables, a change
in an independent variable produces a corresponding change in the dependent variable.
If f is a differentiable function of a single independent variable x,
f f (x)
(1.2-1)
then an infinitesimal change in x given by dx (the differential of x) produces a change
in f given by
df
df
dx
dx
(1.2-2)
where df/dx represents the derivative of f with respect to x and where df represents the
differential of the dependent variable f . The derivative df/dx gives the rate of change
of f with respect to x and is defined by
df
dx
lim
h→0
f (x + h) − f (x)
h
(1.2-3)
if the limit exists. If the derivative exists, the function is said to be differentiable.
There are standard formulas for the derivatives of many functions. For example, if
f a sin(bx), where a and b represent constants, then
df
dx
ab cos(x)
(1.2-4)
If a function depends on several independent variables, each independent variable
makes a contribution like that in Eq. (1.2-2). If f is a differentiable function of x, y,
and z, and if infinitesimal changes dx, dy, and dz are imposed, then the differential df
is given by
1 The Behavior of Gases and Liquids
(the mass of the system) are extensive variables, whereas P and T are intensive
variables. The quotient of two extensive variables is an intensive variable. The density ρ is defined as m/V , and the molar volume V m is defined to equal V /n. These are
intensive variables. One test to determine whether a variable is extensive or intensive
is to imagine combining two identical systems, keeping P and T fixed. Any variable
that has twice the value for the combined system as for one of the original systems
is extensive, and any variable that has the same value is intensive. In later chapters
we will define a number of extensive thermodynamic variables, such as the internal
energy U, the enthalpy H, the entropy S, and the Gibbs energy G.
We are sometimes faced with systems that are not at equilibrium, and the description
of their states is more complicated. However, there are some nonequilibrium states that
we can treat as though they were equilibrium states. For example, if liquid water at
atmospheric pressure is carefully cooled below 0 ◦ C in a smooth container it can remain
in the liquid form for a relatively long time. The water is said to be in a metastable
state. At ordinary pressures, carbon in the form of diamond is in a metastable state,
because it spontaneously tends to convert to graphite (although very slowly).
Differential Calculus and State Variables
Because a dependent variable depends on one or more independent variables, a change
in an independent variable produces a corresponding change in the dependent variable.
If f is a differentiable function of a single independent variable x,
f f (x)
(1.2-1)
then an infinitesimal change in x given by dx (the differential of x) produces a change
in f given by
df
df
dx
dx
(1.2-2)
where df/dx represents the derivative of f with respect to x and where df represents the
differential of the dependent variable f . The derivative df/dx gives the rate of change
of f with respect to x and is defined by
df
dx
lim
h→0
f (x + h) − f (x)
h
(1.2-3)
if the limit exists. If the derivative exists, the function is said to be differentiable.
There are standard formulas for the derivatives of many functions. For example, if
f a sin(bx), where a and b represent constants, then
df
dx
ab cos(x)
(1.2-4)
If a function depends on several independent variables, each independent variable
makes a contribution like that in Eq. (1.2-2). If f is a differentiable function of x, y,
and z, and if infinitesimal changes dx, dy, and dz are imposed, then the differential df
is given by