Equivalence of Kelvin Plank’s and Clausius statement

Although both the definitions of 2nd Law Thermodynamics seems to be different, one working for reversible heat engines and other working for reversed heat engines. But both the definition goes parallel to each other. In other words we can say that violation of 1st statement leads to the violation of 2nd statement.

Violation of Kelvin Plank’s statement leads to violation of Clausius statement.

T₁ > T₂

Let us consider a heat engine which takes Q₁ amount of heat from high temperature source and convert that fully into work done without rejecting heat to the low temperature sink. Therefore it is a violation of Kelvin plank’s statement.

Figure - 1

Let us consider another reversed heat engine working between same temperature limits which takes Q₂ amount of heat from low temperature sink and reject Q₁ + Q₂ amount of heat to high temperature source with help of external work Q₁. The reversed heat engine is working in accordance with Clausius statement. Now as both the devices are working between same temperature limits and as the output of hypothetical heat engine is equal to input of reversed heat engine. Therefore both these devices can be connected to each other and equivalent system is shown in figure 2.

Figure - 2

From this system it is clear that Q₁ amount of heat does not perform any function in the combined system as after leaving from high temperature source it again come back to same source.Therefore this heat may be neglected for further consideration.

The final system is shown in figure 3. Which can be considered as reversed heat engine taking Q₂ amount of heat from low temperature source and reject the same heat to the high temperature source without the help of any external work, thus violates the Clausius statement.

Figure - 3

Violation of Clausius statement leads to violation of Kelvin Plank’s statement.

T₁ > T₂

Let us consider a reverse heat engine which takes Q₂ amount of heat from low temperature sink and reject the same heat to the high temperature source without any work input. Therefore it is violation of Clausius statement.

Figure - 1

Let us consider another heat engine which takes Q₁ amount of heat from high temperature source and produces Q₁ - Q₂ amount of work by rejecting Q₂ amount of heat to low temperature sink. Therefore working in accordance with Kelvin Plank’s statement.

As both the device s are working between same temperature limits so the equivalent system is drawn in figure 2. From this figure it is clear that Q₂ amount of heat rejected by heat engine becomes input for reversed heat engine. Therefore this heat can be neglected for further consideration.

Figure - 2

The final system is shown in figure 3. Which becomes a system that takes Q₁ - Q₂ amount of heat from high temperature source and convert the same heat into work output without rejecting any heat to low temperature sink. Thus violates the Kelvin Plank’s statement.

Figure - 3