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9 Bioenergetics
temperature is desired. When no more thermal energy is transferred, the gas
and the material have reached the same temperature. Of course, the gas volume
must be small enough to not significantly change the temperature of the material
being observed. Using the ideal gas law in the form T = pV /nR, the material
temperature is determined from the gas pressure. In fact, in the basement of
some hospitals, there use to be a thermometer calibration unit, which measured
the pressure of a dilute inert gas within a known volume while a more portable
thermometer needing calibration is inserted in the gas.
4. For a wide range of temperatures, the internal energy E stored in an ideal
monatomic gas made from N monatomic molecules is purely kinetic. According
to the kinetic theory for such gases,
pV = (2/3)N
(1/2)mv
2
,
so E = N
(1/2)mv 2 = (3/2)Nk B T . Thus, for such a gas, temperature
is a measure of the average kinetic energy of the molecules. This does NOT
mean that the temperature of your tongue, or any other system, represents the
average kinetic energy in that system! This is a very common misconception.
The statement connecting temperature to average kinetic energy applies ONLY
to systems which have no intermolecular interaction, such as an ideal gas. When
you put a thermometer on your tongue, the thermal energy transferred to the
thermometer comes from BOTH kinetic energy and potential energy in the
molecules of your tongue.
5. There is another variation on the definition of temperature. From the theoretical
side, it is the cleanest definition: It does not depend on any of the properties of
any particular material. However, it is the least intuitive. Temperature is the rate
at which the internal energy of a system changes per unit change in the disorder
of that system while the volume and particle number of the system are held fixed:
T =
∂E
∂S
V ,N
.
(9.2)
Because the volume is held fixed during the change of the internal energy, no
work is done by or to the system. To interpret this definition, we’ll have to wait
until disorder is associated with a measurable quantity.
Related to this last form of the definition of temperature is the observation that
the quantity S 2 − S 1 =
2
1 δQ/T , a sum of small quantities of heat into a given
system divided by the temperature of that system, taken over a reversible path, 8
8 A thermodynamically ‘reversible process’ is a sequence of small changes in a thermodynamic
system that keeps the system near equilibrium for the whole process and can be taken along the
same sequence of thermodynamic states, but in reverse order, to carry the system back to its initial
condition.
9 Bioenergetics
temperature is desired. When no more thermal energy is transferred, the gas
and the material have reached the same temperature. Of course, the gas volume
must be small enough to not significantly change the temperature of the material
being observed. Using the ideal gas law in the form T = pV /nR, the material
temperature is determined from the gas pressure. In fact, in the basement of
some hospitals, there use to be a thermometer calibration unit, which measured
the pressure of a dilute inert gas within a known volume while a more portable
thermometer needing calibration is inserted in the gas.
4. For a wide range of temperatures, the internal energy E stored in an ideal
monatomic gas made from N monatomic molecules is purely kinetic. According
to the kinetic theory for such gases,
pV = (2/3)N
(1/2)mv
2
,
so E = N
(1/2)mv 2 = (3/2)Nk B T . Thus, for such a gas, temperature
is a measure of the average kinetic energy of the molecules. This does NOT
mean that the temperature of your tongue, or any other system, represents the
average kinetic energy in that system! This is a very common misconception.
The statement connecting temperature to average kinetic energy applies ONLY
to systems which have no intermolecular interaction, such as an ideal gas. When
you put a thermometer on your tongue, the thermal energy transferred to the
thermometer comes from BOTH kinetic energy and potential energy in the
molecules of your tongue.
5. There is another variation on the definition of temperature. From the theoretical
side, it is the cleanest definition: It does not depend on any of the properties of
any particular material. However, it is the least intuitive. Temperature is the rate
at which the internal energy of a system changes per unit change in the disorder
of that system while the volume and particle number of the system are held fixed:
T =
∂E
∂S
V ,N
.
(9.2)
Because the volume is held fixed during the change of the internal energy, no
work is done by or to the system. To interpret this definition, we’ll have to wait
until disorder is associated with a measurable quantity.
Related to this last form of the definition of temperature is the observation that
the quantity S 2 − S 1 =
2
1 δQ/T , a sum of small quantities of heat into a given
system divided by the temperature of that system, taken over a reversible path, 8
8 A thermodynamically ‘reversible process’ is a sequence of small changes in a thermodynamic
system that keeps the system near equilibrium for the whole process and can be taken along the
same sequence of thermodynamic states, but in reverse order, to carry the system back to its initial
condition.