1.2 Systems and States in Physical Chemistry
17
where ≈ means “is approximately equal to” and where we use the common notation
∆V V (final) − V (initial)
(1.2-13)
and so on. Calculations made with Eq. (1.2-12) will usually be more nearly correct if the
finite increments ∆T , ∆P, and ∆n are small, and less nearly correct if the increments
are large.
Variables Related to Partial Derivatives
The isothermal compressibility κ T is defined by
κ T −
1
V
∂V
∂P
(definition of the
isothermal compressibility)
(1.2-14)
The factor 1/V is included so that the compressibility is an intensive variable. The fact
that T and n are fixed in the differentiation means that measurements of the isothermal compressibility are made on a closed system at constant temperature. It is found
experimentally that the compressibility of any system is positive. That is, every system
decreases its volume when the pressure on it is increased.
The coefficient of thermal expansion is defined by
α
1
V
∂V
∂P
P,n
(definition of the coefficient
of thermal expansion)
(1.2-15)
The coefficient of thermal expansion is an intensive quantity and is usually positive.
That is, if the temperature is raised the volume usually increases. There are a few
systems with negative values of the coefficient of thermal expansion. For example,
liquid water has a negative value of α between 0 ◦ C and 3.98 ◦ C. In this range of
temperature the volume of a sample of water decreases if the temperature is raised.
Values of the isothermal compressibility for a few pure liquids at several temperatures
and at two different pressures are given in Table A.1 of Appendix A. The values of the
coefficient of thermal expansion for several substances are listed in Table A.2. Each
value applies only to a single temperature and a single pressure, but the dependence on
temperature and pressure is not large for typical liquids, and these values can usually
be used over fairly wide ranges of temperature and pressure.
For a closed system (constant n) Eq. (1.2-12) can be written
∆V ≈ V α∆T − V κ T ∆P
(1.2-16)
E X A M P L E 1.5
The isothermal compressibility of liquid water at 298.15 K and 1.000 atm is equal to
4.57 × 10 −5 bar −1 4.57 × 10 −10 Pa −1 . Find the fractional change in the volume of a
sample of water if its pressure is changed from 1.000 bar to 50.000 bar at a constant temperature of 298.15 K.
Solution
The compressibility is relatively small in magnitude so we can use Eq. (1.2-16):
∆V ≈ −V κ T ∆P
(1.2-17)
17
where ≈ means “is approximately equal to” and where we use the common notation
∆V V (final) − V (initial)
(1.2-13)
and so on. Calculations made with Eq. (1.2-12) will usually be more nearly correct if the
finite increments ∆T , ∆P, and ∆n are small, and less nearly correct if the increments
are large.
Variables Related to Partial Derivatives
The isothermal compressibility κ T is defined by
κ T −
1
V
∂V
∂P
(definition of the
isothermal compressibility)
(1.2-14)
The factor 1/V is included so that the compressibility is an intensive variable. The fact
that T and n are fixed in the differentiation means that measurements of the isothermal compressibility are made on a closed system at constant temperature. It is found
experimentally that the compressibility of any system is positive. That is, every system
decreases its volume when the pressure on it is increased.
The coefficient of thermal expansion is defined by
α
1
V
∂V
∂P
P,n
(definition of the coefficient
of thermal expansion)
(1.2-15)
The coefficient of thermal expansion is an intensive quantity and is usually positive.
That is, if the temperature is raised the volume usually increases. There are a few
systems with negative values of the coefficient of thermal expansion. For example,
liquid water has a negative value of α between 0 ◦ C and 3.98 ◦ C. In this range of
temperature the volume of a sample of water decreases if the temperature is raised.
Values of the isothermal compressibility for a few pure liquids at several temperatures
and at two different pressures are given in Table A.1 of Appendix A. The values of the
coefficient of thermal expansion for several substances are listed in Table A.2. Each
value applies only to a single temperature and a single pressure, but the dependence on
temperature and pressure is not large for typical liquids, and these values can usually
be used over fairly wide ranges of temperature and pressure.
For a closed system (constant n) Eq. (1.2-12) can be written
∆V ≈ V α∆T − V κ T ∆P
(1.2-16)
E X A M P L E 1.5
The isothermal compressibility of liquid water at 298.15 K and 1.000 atm is equal to
4.57 × 10 −5 bar −1 4.57 × 10 −10 Pa −1 . Find the fractional change in the volume of a
sample of water if its pressure is changed from 1.000 bar to 50.000 bar at a constant temperature of 298.15 K.
Solution
The compressibility is relatively small in magnitude so we can use Eq. (1.2-16):
∆V ≈ −V κ T ∆P
(1.2-17)