Clausius theorem

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Thermodynamics
Carnot heat engine 2.svg
Equations
Carnot's theorem
Clausius theorem
Fundamental relation
Ideal gas law
Maxwell relations
Table of thermodynamic equations

Overview

The Clausius theorem (1855) states that in a cyclic process

\oint \frac{\delta Q}{T} \leq 0,

where δQ is the amount of heat absorbed. The equality holds in the reversible case[1] and the '<' is in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic process the variation of a state function is zero.

History

The Clausius Theorem is a mathematical explanation of the Second Law of Thermodynamics. Also referred to as the “Inequality of Clausius”, the theorem was developed by Rudolph Clausius and intends to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius Theorem provides a quantitative formula for understanding the second law.

Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius Theorem was first published in 1862 in Clausius’ sixth memoir, “On the Application of the Theorem of the Equivalence of Transformations to Interior Work”. Clausius sought to show a proportional relationship between entropy and the heat flow (dQ) through a system. In a system, heat can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that “The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.” In other words, the equation

\oint \frac{\delta Q}{T} = 0

where δQ is heat flow given up by the body to a reservoir (note the different sign convention from above) and T is absolute temperature of the body when this heat is given up) must be found to be true for any process that is cyclical and reversible. He then took this a step further and determined that the following equation must be found true for any cyclical process that is possible, reversible or not. This equation is the “Clausius Inequality”.

\oint \frac{\delta Q}{T} \leq 0

Now that this is known, there must be a relation developed between the Clausius Inequality and entropy. The change in entropy S is defined as

\Delta S {{=}} \int \frac{\delta Q}{T}

Using all of equations above, we see that the change in entropy gives us insight into whether or not a cyclical system in reversible or irreversible. In any thermodynamic process, you can have both positive and negative transformations of heat and work. For a reversible process, the positive and negative transformations must balance out and equal zero. For irreversible processes, the positive and negative transformations do not necessarily have to equal zero, but the positive changes must outweigh the negative changes. If the change in entropy is greater than zero, then the process is irreversible. If the change in entropy is equal to zero, then the process is reversible. A negative change in entropy is not possible. This directly ties into the Second Law of Thermodynamics, which states that “The entropy of the Universe is increasing”. It must be emphasized that the entropy of a system is a state function. This means that it depends only on what state the system is currently in, and not what path the system took to get there.

Proof

Proving Clausius Inequality

Suppose a system absorbs heat \delta Q at temperature T. Since the value of \frac{\delta Q}{T} does not depend on the details of how the heat is transferred, we can assume it is from a Carnot engine, which in turn absorbs heat \delta Q_0 from a heat reservoir with constant temperature T_0.

According to the nature of Carnot cycle,

\frac{\delta Q}{T}=\frac{\delta Q_0}{T_0}
\Rightarrow\delta Q_0=T_0\frac{\delta Q}{T}

Therefore in one cycle, the total heat absorbed from the reservoir is

Q_0=T_0\oint\frac{\delta Q}{T}

Since after a cycle, the system and the Carnot engine as a whole return to its initial status, the difference of the internal energy is zero. Thus according to First Law of Thermodynamics,

Q_0=\Delta U+W+W_0=W+W_0=W_\text{total}

According to the Kelvin statement of Second Law of thermodynamics, we cannot drain heat from one reservoir and convert them entirely into work without making any other changes, so

W_\text{total}\leq 0

Combine all the above and we get Clausius inequality

\oint\frac{\delta Q}{T}\leq 0

If the system is reversible, then reverse its path and do the experiment again we can get

-\oint\frac{\delta Q}{T}\leq 0

Thus

\oint\frac{\delta Q}{T}=0

See also

References

  1. ^ Clausius theorem at Wolfram Research
  • Morton, A. S., and P.J. Beckett. Basic Thermodynamics. New York: Philosophical Library Inc., 1969. Print.
  • Saad, Michel A. Thermodynamics for Engineers. Englewood Cliffs: Prentice-Hall, 1966. Print.
  • Hsieh, Jui Sheng. Principles of Thermodynamics. Washington, D.C.: Scripta Book Company, 1975. Print.
  • Lemansky, Mark W. Heat and Thermodynamics. 4th ed. New York: McGwaw-Hill Book Company, 1957. Print.
  • Clausius, Rudolph. The Mechanical Theory of Heat. London: Taylor and Francis, 1867. eBook


External links