The entropy change of the universe is the sum of the entropy change of the system and the entropy change of the surroundings:
ΔSuniverse
= ΔSsystem + ΔSsurrounding
To determine the entropy change of the universe, let us consider the following processes:
- Adiabatic expansion
- Adiabatic compression
- Heat Engine
- Heat Pump
Adiabatic expansion
Let us consider a reversible adiabatic expansion of a gas in a piston-cylinder device from P1 to P2. Since work is done by the system, the gas temperature decreases to the final state temperature T2. The process is represented on a P – T diagram below:
During expansion, work is done by the system and -W
is a positive quantity. As the initial and final state pressures are the same,
part of the work done by the system during the irreversible expansion process
is used to overcome friction and as a result:
This indicates that the irreversible process leads to
a smaller decrease in internal energy than the reversible process. Equation 3
can be expressed as:
U1
– U2 > U1 – U3
thus U3 > U2 (4)
Therefore T3
> T2. It is possible to go from state 3 to state 2 by
removing heat reversibly to decrease the temperature. The change in entropy for
this process is given by:
Since heat is removed
from the system, i.e., dQ < 0, then S3 > S2.
On the other hand, the
reversible process occurs at constant entropy, (S1 = S2),
and thus S3 > S1. Note that the processes 1 → 2 and 1
→ 3 cause no entropy change of the surroundings as they both are adiabatic.
Hence, the system entropy change is the entropy change of the universe. The universe
entropy changes due to the reversible and irreversible adiabatic expansion
processes are summarized as follows:
Adiabatic Compression
Let us consider a reversible adiabatic compression of
a gas in a piston-cylinder device from P1 to P2. Since
work is done on the system, the gas temperature will increase to the final
state temperature, T2. The process is represented on a P – T diagram
as follows:
During compression, work is done on the system and W
is positive. Since the initial and final state pressures are the same, the
irreversible compression requires an extra work to overcome friction and as a
result:
This indicates that the irreversible process leads to
a larger increase in internal energy than the reversible process. Equation 8
can then be expressed as:
U3
– U1 > U2 – U1
thus U3 > U2 (9)
Therefore T3 > T2. It is possible to go from state 3 to state 2 by removing heat reversibly to decrease the temperature. The change in entropy for this process is given by:
Since heat is removed
from the system, i.e., dQ < 0, then S3 > S2.
On the other hand, the
reversible process occurs at constant entropy, (S1 = S2),
and thus S3 > S1. These processes have no entropy change
of the surroundings as they both are adiabatic. Hence, the system entropy
change is the entropy change of the universe. The universe entropy changes due
to the reversible and irreversible adiabatic expansion processes are summarized
as follows:
Heat Engine
Let us consider two heat
engines that operate between heat reservoirs at the same constant temperatures TH
and TC as shown below:
The reversible on
extracts QH amount of heat from the high temperature reservoir and
discards Qcrev amount of heat to the low temperature reservoir, with
a work output of -Wrev.
The irreversible heat engine
receives the same amount of heat from the high temperature reservoir and discards
Qcirrev amount of heat to the low temperature reservoir, with a work
output of − Wirrev.
The entropy change of the
universe is:
ΔSuniverse = ΔSH + ΔSheat engine + ΔSC (11)
As the heat engine
operates is cyclical, there is no entropy change of the heat engine and thus Eq.
11 simplifies as:
ΔSuniverse
= ΔSH + ΔSC (12)
For a reversible heat engine, equation 12 can be expressed as:
For a heat engine ΔU = 0
and − W = Q. Thus,
QH
- Qcrev
= -Wrev and QH - Qcirrev
= -Wirrev
(17)
Since both of the heat
engines operate between the same temperature limits, the work output of the reversible
heat engine is always greater than the irreversible one, i.e.,
-Wrev
> -Wirrev (18)
Using equation 18 in
equation 17 we see that:
Qcirrev > Qcrev
(19)
Using the identity in equation
19, we can compares equations 15 and 16 to result in:
ΔSuniverse|irrev
> 0 (20)
Heat Pump
Let us consider two heat pumps (refrigerators) that operate between the same constant temperature heat reservoirs TH and TC as shown:
The reversible heat pump
extracts QC amount of heat from the low temperature reservoir and
discards QHrev amount of heat to the high temperature reservoir. Thus,
Wrev amount of work is done on the heat pump.
The irreversible heat pump
extracts the same amount of heat from the low temperature reservoir but
discards QHirrev amount of heat to the high temperature reservoir
while receiving Wirrev amount of work from the surroundings.
The entropy change of the universe is expressed as:
ΔSuniverse = ΔSH + ΔSheat pump + ΔSC (21)
As mentioned previously, since the heat engine operates is cyclical, there is no entropy change of the heat engine and thus Eq. 11 simplifies as:
ΔSuniverse = ΔSH + ΔSC (22)
For a reversible heat pump, equation 22 can be expressed as:
For a heat pump ΔU = 0 and − W = Q. Thus,
QC – QHrev = -Wrev and QC – QHirrev = -Wirrev (26)
Since both of the heat engines operate between the same temperature limits, the work received by the reversible heat engine is always less than the irreversible one, i.e.,
Wrev < Wirrev (27)
Using equation 27 in equation 26 we see that:
QHirrev >
QHrev (28)
Since
QHirrev is larger than in the reversible case, QHrev,
comparing Equations 24 and 25 results in:
ΔSuniverse|irrev
> 0 (29)
Note:
The results for the thermal processes (Heat engine & heat pump) are the same as those obtained for the mechanical processes (Adiabatic expansion & compression) and can be summarized by the equation
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