48
2 Work, Heat, and Energy: The First Law of Thermodynamics
Exact and Inexact Differentials
The differential of a function of two or more independent variables is called an exact
differential. If f is a function of x and y, the differential
df
∂f
∂x
y
dx +
∂f
∂y
x
dx (exact differential)
(2.1-19)
is an exact differential. There is an important theorem of mathematics concerning
the line integral of an exact differential: A line integral of an exact differential is
equal to the function evaluated at the final point of the integration curve minus the
function evaluated at the initial point of the curve. The line integral of an exact
differential therefore depends only on the end points, and not on the curve connecting them. The line integral of an exact differential is said to be path-independent.
The converse of this theorem is also true. If the integral of a differential is pathindependent for all paths between the same end points, the differential must be an exact
differential.
Since the pressure P is a state function, dP is an exact differential. The line integral
of dP in part b of Example 2.2 is equal to the value of P at the end of the process minus
the value of P at the beginning of the process:
c
dP ∆P P(final) − P(initial)
Exercise 2.3
Verify that ∆P in Example 2.2 is equal to the final pressure minus the initial pressure.
Work Is an Inexact Differential
A differential that is not exact is called an inexact differential. The differential
du M(x, y)dx + N(x, y)dy
(2.1-20)
is an inexact differential if M and N are not the appropriate partial derivatives of
the same function. The line integral of an inexact differential depends on the path of
integration as well as on the initial point and the final point. We will show that dw is an
inexact differential by showing that two processes with the same initial and final states
can correspond to different amounts of work done on the system.
E X A M P L E 2.7
Consider a reversible process with the same initial and final states as the process of
Example 2.2, but with a different path. Calculate the work done on the ideal gas system
of Example 2.2 if it is reversibly cooled at constant volume of 5.000 L from 298.15 K
2 Work, Heat, and Energy: The First Law of Thermodynamics
Exact and Inexact Differentials
The differential of a function of two or more independent variables is called an exact
differential. If f is a function of x and y, the differential
df
∂f
∂x
y
dx +
∂f
∂y
x
dx (exact differential)
(2.1-19)
is an exact differential. There is an important theorem of mathematics concerning
the line integral of an exact differential: A line integral of an exact differential is
equal to the function evaluated at the final point of the integration curve minus the
function evaluated at the initial point of the curve. The line integral of an exact
differential therefore depends only on the end points, and not on the curve connecting them. The line integral of an exact differential is said to be path-independent.
The converse of this theorem is also true. If the integral of a differential is pathindependent for all paths between the same end points, the differential must be an exact
differential.
Since the pressure P is a state function, dP is an exact differential. The line integral
of dP in part b of Example 2.2 is equal to the value of P at the end of the process minus
the value of P at the beginning of the process:
c
dP ∆P P(final) − P(initial)
Exercise 2.3
Verify that ∆P in Example 2.2 is equal to the final pressure minus the initial pressure.
Work Is an Inexact Differential
A differential that is not exact is called an inexact differential. The differential
du M(x, y)dx + N(x, y)dy
(2.1-20)
is an inexact differential if M and N are not the appropriate partial derivatives of
the same function. The line integral of an inexact differential depends on the path of
integration as well as on the initial point and the final point. We will show that dw is an
inexact differential by showing that two processes with the same initial and final states
can correspond to different amounts of work done on the system.
E X A M P L E 2.7
Consider a reversible process with the same initial and final states as the process of
Example 2.2, but with a different path. Calculate the work done on the ideal gas system
of Example 2.2 if it is reversibly cooled at constant volume of 5.000 L from 298.15 K