Clausius Statement

It is nearly impossible for heat to move by itself from a temperature that is lower in temperature to a reservoir that is at a higher temperature. That is we can say that the transfer of heat can only occur spontaneously from high temperature to temperature. For example, we cannot construct a refrigerator that can operate without any input work.

That is, the transfer of heat can only occur spontaneously in the temperature direction which decreases. 

Clausius Theorem

The theorem of Clausius in 1855 states that for a thermodynamic system, that is the heat engine or even heat pump exchanging heat with external reservoirs and thermodynamics undergoing of a cycle follows the inequality given by

Statement Which is of Celcius 

This is the amount of infinitesimal heat which is absorbed by the system from the reservoir and is the external temperature of a reservoir that is surrounded at a particular instant in time. The integral which is closed is carried out along a process of thermodynamic process path from the initial/final state to the same initial/final state. In principle, the closed integral can start and end at a point that is arbitrary along the path.

If there are multiple reservoirs with different temperatures then the inequality of Clausius reads the following:

In the case of a process that is reversible, then it holds equality. The case which is reversible is used to introduce the function of the state which is known as entropy. This is because in a process that is cyclic the variation of a state function is zero. In other words, we can say that the Clausius statement says that it is impossible to construct a device whose only effect is the transfer of heat from a reservoir that is cool to a hot reservoir. Equivalently the spontaneously flows of heat from a hot body to a cooler one.

The generalization is Clausius inequality.

History of Clausius theorem 

The theorem of Clausius is an explanation of mathematics of the thermodynamics second law. It was developed by Rudolf Clausius who explained the relationship between the flow of heat flow in a system and the entropy of the system and its surroundings. The development of Clausius in his efforts to explain entropy is quantitatively defined. In items that are direct terms, the theorem gives us a way to determine if a process of cyclical is irreversible or reversible. The theorem of Clausius provides a formula that is quantitative for understanding the second law.

Clausius was one of the first to work on the idea of entropy and he is even responsible for giving that name to the theorem. What is now known as the theorem of Clausius was first published in 1862 in Clausius' ``On the Application of the Theorem of the transformation Equivalence to Interior Work". Sought Clausius to show a proportional relationship which is between entropy and the energy flow by heating defined by the symbol δQ into a system. 

In a system, the energy of heat can be transformed into work, and the work can be transformed into heat through a process that is cyclical. Clausius writes that The algebraic sum of all the transformations which are occurring in a process that is cyclical can only be less than zero, or, as a case of extreme equal to nothing. In other words, we can say that the equation is given below:

Clause Statement of Second Law of Thermodynamics

In 1850 Here we can see that one of the earliest statements of the thermodynamics Second Law was made by R. Clausius. He stated the following in the next line.

“It is very impossible to construct a device that operates on a cycle and whose effect is the heat transfer from a body which was cooler to a hotter one. 

Heat spontaneously cannot flow from the system of cold to the hot system without external work being performed on the system. This is exactly what heat pumps and refrigerators accomplish. In a heat flow or the refrigerator from the body which is cold to hot, but only when forced by an external work, the refrigerators are driven by motors which are electric and require work from their surroundings to operate.

The Clausius statement of law states the following:

It is nearly impossible for any system to operate in a way such that the sole result would be a transfer of energy by heat from a cooler to a hotter body.

Heat can be transferred from a body that is cooler to a body that is hotter if other effects accomplishing the heat transfer occur within the surrounding system or both. Air conditioners or AC and refrigerators are devices that are supposed to transfer heat from a cool space to its hot surroundings. But in both cases, they need power input. The statement which is of Clausius says that an air conditioner won't cool a room without power input.

Kelvin-Planck Statement of the Second Law

An engine or a heat engine must reject some energy to sink the heat to run a cycle. That is no engine of heat that can convert all the heat it received from a temperature of the high source to work. It is the basis for the Plancks of Kelvin statement of the thermodynamics second law which is given as:

Conclusion

This is all about the Clausius Statement explaining thermodynamics and its related terms. Focus on the history of this statement and how it defines the laws of thermodynamics to grab hold of the concept.