34
2 Macroscopic Thermodynamics
with U the change in the internal energy for the process as a whole, while Q and
W are the heat exchanged and the work done during the process as a whole.
Because it is possible both for a thermodynamic system to perform work on
its surroundings (which we shall refer to as work done by the system) and for its
surroundings to do work on a system, we shall require a sign convention for W . For
the First Law as expressed by Eq. (2.2.4b), the convention is that W represents the
work done on the system by its surroundings, so that a negative value for W signifies
that work has been done by the system on its surroundings. Similarly, as energy can
be transferred as ‘heat’ either to the thermodynamic system from its surroundings
or from the system to its surroundings, we need to be clear on the sign convention
for Q: the traditional convention (herein adopted) is that Q is positive when energy
is transferred to the system from its surroundings, corresponding to an increase in
the energy of the system.
Dimensionally, it will be clear that the product P V represents pressure–volume
work, as pressure is a force per unit area, so that P V has dimensions of force times
distance, i.e., work. The incremental work done by (reversibly) changing the volume
of a gaseous thermodynamic system by an amount dV will thus be represented by
−P dV according to the sign convention that has been adopted herein: the minus
sign ensures that work is done on the system by the surroundings in the compression
of a gas, for which dV < 0. The pressure–volume work done in changing the
system volume from an initial value V i to a final value V f is thus given by
W = −
V f
V i
P dV ,
(2.2.5)
with the pressure, P , determined from the equation of state. For example, for an
ideal classical gas, the pressure is given by P = Nk B T /V , so that the work done
on the system in carrying out an isothermal compression from volume V i to volume
V f (V f < V i ) is
W = −Nk B T
V f
V i
dV
V
= −Nk B T ln
V f
V i
> 0 ,
while the work done by the system in an isothermal expansion (V f > V i ) will be
W = −Nk B T ln
V f
V i
< 0 .
As the internal energy of an ideal classical gas depends only upon the temperature, U will be zero for any isothermal change, so that the energy transferred
(as heat) between the system surroundings (also referred to as the reservoir) and the
thermodynamic system will be the negative of the system work. Hence, compression
of an ideal classical gas necessitates, via the First Law, that Q(system) be negative,
corresponding to energy transferred (as heat) to the surroundings.
2 Macroscopic Thermodynamics
with U the change in the internal energy for the process as a whole, while Q and
W are the heat exchanged and the work done during the process as a whole.
Because it is possible both for a thermodynamic system to perform work on
its surroundings (which we shall refer to as work done by the system) and for its
surroundings to do work on a system, we shall require a sign convention for W . For
the First Law as expressed by Eq. (2.2.4b), the convention is that W represents the
work done on the system by its surroundings, so that a negative value for W signifies
that work has been done by the system on its surroundings. Similarly, as energy can
be transferred as ‘heat’ either to the thermodynamic system from its surroundings
or from the system to its surroundings, we need to be clear on the sign convention
for Q: the traditional convention (herein adopted) is that Q is positive when energy
is transferred to the system from its surroundings, corresponding to an increase in
the energy of the system.
Dimensionally, it will be clear that the product P V represents pressure–volume
work, as pressure is a force per unit area, so that P V has dimensions of force times
distance, i.e., work. The incremental work done by (reversibly) changing the volume
of a gaseous thermodynamic system by an amount dV will thus be represented by
−P dV according to the sign convention that has been adopted herein: the minus
sign ensures that work is done on the system by the surroundings in the compression
of a gas, for which dV < 0. The pressure–volume work done in changing the
system volume from an initial value V i to a final value V f is thus given by
W = −
V f
V i
P dV ,
(2.2.5)
with the pressure, P , determined from the equation of state. For example, for an
ideal classical gas, the pressure is given by P = Nk B T /V , so that the work done
on the system in carrying out an isothermal compression from volume V i to volume
V f (V f < V i ) is
W = −Nk B T
V f
V i
dV
V
= −Nk B T ln
V f
V i
> 0 ,
while the work done by the system in an isothermal expansion (V f > V i ) will be
W = −Nk B T ln
V f
V i
< 0 .
As the internal energy of an ideal classical gas depends only upon the temperature, U will be zero for any isothermal change, so that the energy transferred
(as heat) between the system surroundings (also referred to as the reservoir) and the
thermodynamic system will be the negative of the system work. Hence, compression
of an ideal classical gas necessitates, via the First Law, that Q(system) be negative,
corresponding to energy transferred (as heat) to the surroundings.