Equivalence
of Kelvin-Planck and Clausius Statements
The Clausius and Kelvin-Planck statements of the second law are
entirely equivalent. This equivalence can be demonstrated by showing that the
violation of either statement can result in violation of the other one.
Referring to Figure 4.4(a)
the device marked Clausius violator is pumping Q1 amount of heat from a low temperature
reservoir at T1 to a high temperature
reservoir at T2 without the aid of any
external agency. This is an impossible arrangement.
If such an arrangement is possible it would also violate
Kelvin-Planck statement. Let a heat engine operating between the same
reservoirs at T2 and T1 take in Q2 as heat input at T2. It
converts a part of this heat into work and rejects heat Q3to the sink
at T1. Since the Clausius violator is rejecting the same quantity Q2at T2, it can be
supplied directly into the heat engine so that the reservoir at T2 can be eliminated. This combination as shown
in Figure 4.4 (b) is producing continuous work with a single reservoir at T1. Hence it
violates the Kelvin-Planck statement.
Referring to Figure 4.5 a Kelvin-planck violator is converting all
heat QH taken from the reservoir at
TH into work. If such an impossible heat engine
is assumed to exist it will violate the Clausius statement. Consider a
refrigerator pumping QL heat from the low
temperature reservoir at TL to the reservoir at higher
temperature TH. Combined with the Kelvin-Planck violator, the arrangement is
pumping QL heat from TL to TH, without any external agency.
Hence it violate the Clausius statement.
Reversible
Process
A process is said to be reversible if it can be reversed without
leaving any trace on the surroundings.
For example, let a system be taken from state 1 to state 2 with a
work transfer of +5 kJ and heat transfer of -10 kJ. If the process is reversible, while
taking the system from state 2 to state 1, the work transfer must be -5 kJ and heat transfer must be +10
kJ. So that, both the system and surroundings are returned to their initial
states at the end of the process 2 to 1.
