9.4 Definition of Heat
329
is, in fact, process independent, and only dependent on the endpoints. For this
reason, the inverse temperature has been called the ‘integrating factor of heat’.
For each of the definitions, a common scale for temperature is introduced.
Looking from our current perspective, the most natural scale for temperature would
have been to adopt energy units. However, to communicate with our fellow humans,
we use the Celsius scale 9 for which, at first, the freezing point and boiling point of
water were set at 0 and 100 ◦ C measured at one atmosphere, but then absolute zero
was set at −273.15 ◦ C and the triple point of water, was set at 0.01 ◦ C. The Kelvin
unit is the same as the Celsius unit, but starts with 0 ◦ at absolute zero. Conversion
to energy units is effected by multiplying with Boltzmann’s constant
k B ≡ 1.380649 × 10
−23 J/deg.
9.4 Definition of Heat
The energy transferred between two macrosystems due to temperature differences
is heat. Through the definition of temperature, heat always goes from a hotter to a
colder body. The fact that a temperature can be identified for each of the systems
implies that in this definition, each system is at or near thermal equilibrium while
heat is being transferred. Conventionally, heat which goes into a given system is
taken as positive, while heat which goes out is taken as negative.
The internal energy of any macrosystem can always be expressed in terms of
the energies of the microstates, as shown in Eq. (9.1). Small changes in the internal
energy of the system can occur by energy transfer into or out of the system. From
the expression for E, a small energy transfer can be divided into two distinct terms:
One caused by small changes in the numbers of entities n i (such as particles) in the
energy states with energy i , and the second by changes in the energies i of the
states of the system:
dE =
i
dn i i +
i
n i dd i .
(9.3)
Since the n i are integers, a calculus notation dn i requires explanation. Most of the
n i for macrosystems near equilibrium are typically astronomically large, so dn i can
be integer and still be much smaller than n i . i.e. infinitesimal by comparison. The
microscopic distinction between heat and work is pictorially represented in Fig. 9.1.
The volume dependence of the total energy in a macrosystem comes into the
internal energy expression Eq. (9.1) through the energies i . For example, consider
a box confining a particle bouncing between the walls of the box. In quantum
9 The term Celsius has replaced Centigrade.
329
is, in fact, process independent, and only dependent on the endpoints. For this
reason, the inverse temperature has been called the ‘integrating factor of heat’.
For each of the definitions, a common scale for temperature is introduced.
Looking from our current perspective, the most natural scale for temperature would
have been to adopt energy units. However, to communicate with our fellow humans,
we use the Celsius scale 9 for which, at first, the freezing point and boiling point of
water were set at 0 and 100 ◦ C measured at one atmosphere, but then absolute zero
was set at −273.15 ◦ C and the triple point of water, was set at 0.01 ◦ C. The Kelvin
unit is the same as the Celsius unit, but starts with 0 ◦ at absolute zero. Conversion
to energy units is effected by multiplying with Boltzmann’s constant
k B ≡ 1.380649 × 10
−23 J/deg.
9.4 Definition of Heat
The energy transferred between two macrosystems due to temperature differences
is heat. Through the definition of temperature, heat always goes from a hotter to a
colder body. The fact that a temperature can be identified for each of the systems
implies that in this definition, each system is at or near thermal equilibrium while
heat is being transferred. Conventionally, heat which goes into a given system is
taken as positive, while heat which goes out is taken as negative.
The internal energy of any macrosystem can always be expressed in terms of
the energies of the microstates, as shown in Eq. (9.1). Small changes in the internal
energy of the system can occur by energy transfer into or out of the system. From
the expression for E, a small energy transfer can be divided into two distinct terms:
One caused by small changes in the numbers of entities n i (such as particles) in the
energy states with energy i , and the second by changes in the energies i of the
states of the system:
dE =
i
dn i i +
i
n i dd i .
(9.3)
Since the n i are integers, a calculus notation dn i requires explanation. Most of the
n i for macrosystems near equilibrium are typically astronomically large, so dn i can
be integer and still be much smaller than n i . i.e. infinitesimal by comparison. The
microscopic distinction between heat and work is pictorially represented in Fig. 9.1.
The volume dependence of the total energy in a macrosystem comes into the
internal energy expression Eq. (9.1) through the energies i . For example, consider
a box confining a particle bouncing between the walls of the box. In quantum
9 The term Celsius has replaced Centigrade.