2.3.2 Heat exchange and heat capacity
The flow of heat (designated as q) is a form of energy transfer that results
from a temperature difference between the system and surroundings.
By convention, a minus sign is used (−q) to indicate the transfer of heat
from the system to the surroundings. A positive value of q indicates the
absorption of heat into the system from the surroundings. The heat
capacity of a material is defined as the ratio of the heat transferred to the
temperature change, or more accurately,
C T
ð Þ =
dq
dT
(2.18)
The heat capacity does itself depend on temperature, but is usually
considered constant within a small temperature range. The heat transferred due to a change in temperature is
q =
ð T 2
T 1
C T
ð ÞdT
(2.19)
Heat capacity is both an extensive quantity and a path function. It can be
directly measured, and like any extensive quantity, dividing by the number
of moles gives the corresponding intensive quantity. It is important to
recognize that molar heat capacity is a path function and values depend on
whether they are measured under constant volume (C V ) or constant
pressure (C P ) conditions. For solids and liquids, the difference between the
two is insignificant. However, for gases the difference between C V and C P is
approximately equal to the molar gas constant R (or 8.134 J mol
−1 K
−1
).
2.3.3 The first law in terms of work and heat
Now that we have discussed work and heat, let’s consider processes in which
both are exchanged. For illustrative purposes, let’s consider the expansion of
the gas shown in Figure 2.1. Clearly, the expansion involves work done on
the surroundings. If the process occurs adiabatically (i.e., no heat transfer),
then the gas suffers a drop in internal energy. This drop in internal energy is
manifested by a decrease in the temperature of the gas. Indeed, experience
shows us that gases expanding in a canister feel cool at the nozzle.
In an isothermal expansion, where the temperature of the system remains
constant, heat enters the system to maintain the constant temperature
of the gas. Thus, in a reversible isothermal expansion process, the work
done on the surroundings is exactly equal to the heat transferred into the
CHAPTER 2: Thermodynamics and Nanoscience
36
The flow of heat (designated as q) is a form of energy transfer that results
from a temperature difference between the system and surroundings.
By convention, a minus sign is used (−q) to indicate the transfer of heat
from the system to the surroundings. A positive value of q indicates the
absorption of heat into the system from the surroundings. The heat
capacity of a material is defined as the ratio of the heat transferred to the
temperature change, or more accurately,
C T
ð Þ =
dq
dT
(2.18)
The heat capacity does itself depend on temperature, but is usually
considered constant within a small temperature range. The heat transferred due to a change in temperature is
q =
ð T 2
T 1
C T
ð ÞdT
(2.19)
Heat capacity is both an extensive quantity and a path function. It can be
directly measured, and like any extensive quantity, dividing by the number
of moles gives the corresponding intensive quantity. It is important to
recognize that molar heat capacity is a path function and values depend on
whether they are measured under constant volume (C V ) or constant
pressure (C P ) conditions. For solids and liquids, the difference between the
two is insignificant. However, for gases the difference between C V and C P is
approximately equal to the molar gas constant R (or 8.134 J mol
−1 K
−1
).
2.3.3 The first law in terms of work and heat
Now that we have discussed work and heat, let’s consider processes in which
both are exchanged. For illustrative purposes, let’s consider the expansion of
the gas shown in Figure 2.1. Clearly, the expansion involves work done on
the surroundings. If the process occurs adiabatically (i.e., no heat transfer),
then the gas suffers a drop in internal energy. This drop in internal energy is
manifested by a decrease in the temperature of the gas. Indeed, experience
shows us that gases expanding in a canister feel cool at the nozzle.
In an isothermal expansion, where the temperature of the system remains
constant, heat enters the system to maintain the constant temperature
of the gas. Thus, in a reversible isothermal expansion process, the work
done on the surroundings is exactly equal to the heat transferred into the
CHAPTER 2: Thermodynamics and Nanoscience
36