We were discussing various basic concepts of thermodynamics such as “Steady flow energy equation” in our recent post. We have also discussed the “Flow work or flow energy in thermodynamics and also “Steady flow energy equation for throttling devices” and “steady flow energy equation for heat exchanger” too in the field of thermal engineering.

Today we will see here the very important topic i.e. Equivalence of Kelvin-plank and Clausius statement with the help of this post.

Before understanding the concept of equivalence of Kelvin plank statement and Clausius statement, let us brief here the second law of thermodynamics.

There are two statements of second law of thermodynamics i.e.  Kelvin- plank statement and Clausius statement.

### Clausius statement for second law of thermodynamics

It is not possible for a self acting machine working in a cyclic process unaided by any external agency, for conveying heat energy from an object at lower temperature to an object at higher temperature.

### Kelvin plank statement for second law of thermodynamics

It is not possible to construct one engine, which while operating in a cycle produces no other effect except to extract heat energy from a single thermal energy reservoir and do equivalent amount of work.

For more detailed explanation about second law of thermodynamics please find the link “Second law of thermodynamics”.

### Let us come to the point i.e. Equivalence of Kelvin plank and Clausius statement

As we have seen above both statements of second law of thermodynamics i.e. Kelvin plank and Clausius statement and we may think that these two statements are different with each other.

But if we will analyze these statements, we will conclude that both statements i.e. Kelvin plank statement and Clausius statement of second law of thermodynamics are equivalent with each other in each respect.

Now we will have one question, how we can say that both statements (Kelvin plank statement and Clausius statement) of second law of thermodynamics are parallel statements or equivalent with each other in all respects.

### So let us prove it

We will show here the equivalence of Kelvin plank statement and Clausius statement of second law of thermodynamics by showing that violation of one statement signifies the violation of other statement and vice versa.

Violation of one statement signifies the violation of other statement means if a thermodynamic system is not following the Kelvin plank statement of second law of thermodynamics then it will also not follow the Clausius statement too.

Similarly, if a system is not following the Clausius statement of second law of thermodynamic then it will also not follow the Kelvin plank statement of second law of thermodynamics.

Now we will show that violation of one statement signifies the violation of other statement in order to show that Kelvin plank statement and Clausius statement of second law of thermodynamics are equivalent with each other in each respect.

Let us see the following figure, where a heat pump is transferring heat energy Q1 from a lower temperature body to a higher temperature body without any other effect i.e. heat pump is receiving heat energy Q1 from a lower temperature body and delivering this heat energy Q1 to a higher temperature body without securing any work energy from surrounding.
Hence we can say that clausius statement of second law of thermodynamic is not followed here by the system and clausius statement is violated here.

### Now let us see here that whether Kelvin plank statement is also violated or not for this system

Let us consider one heat engine which is also operating between the same thermal energy reservoirs as shown in figure. Hot thermal energy reservoir has temperature T1 and Cold thermal energy reservoir has temperature T2. Heat engine is receiving heat energy Q1 from hot thermal energy reservoir and producing net work W by rejecting heat energy Q2 to low temperature thermal energy reservoir.

We have considered here that heat engine is receiving same quantity of heat energy which is discharged by heat pump to hot thermal energy reservoir.

Therefore, hot thermal energy reservoir could be eliminated here as discharged quantity of heat energy by the heat pump could be taken as input heat energy by the heat engine.

Now we can see here that heat engine and heat pump consisting one heat engine which is producing work W during working together in a cycle by exchanging heat energy from a single thermal energy reservoir and it shows that this system has not followed the Kelvin plank statement of second law of thermodynamics.

Or we can say that Kelvin plank statement of second law of thermodynamics is violated here for this system and we have shown here that if a system is not following the Clausius statement of second law of thermodynamic then it will also not follow the Kelvin plank statement of second law of thermodynamics.

Similarly, we can show that if a thermodynamic system is not following the Kelvin plank statement of second law of thermodynamics then it will also not follow the Clausius statement too.

So, we have proved here that Kelvin plank statement and Clausius statement of second law of thermodynamics are equivalent with each other in each respect.

Do you have suggestions? Please write in comment box
We will see another topic i.e. "Reversed Carnot heat engine cycle" in our next post in the category of thermal engineering.

### Reference:

Engineering thermodynamics by P.K. Nag