Clausius theorem

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The Clausius theorem (1855) states that for a system exchanging heat with external reservoirs and undergoing a cyclic process (i.e. a process which ultimately returns a system to its original state):

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

where δ$Q$ is the amount of heat absorbed by the system from the reservoir and $T$ is the temperature of the external reservoir at a particular instant. The equality holds in the reversible case[1] and the inequality holds in the irreversible case. The reversible case is used to introduce the entropy state function. 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 Rudolf Clausius who intended 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 energy flow by heating (δQ) into a system. In a system, this heat energy 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 less than zero, or, as an extreme case, equal to nothing.” In other words, the equation

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

where δQ is energy flow into the system due to heating and T is absolute temperature of the body when that energy is absorbed is found to be true for any process that is cyclical and reversible. Clausius 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 amount of entropy S added to the system during the cycle is defined as

$\Delta S {{=}} \oint \frac{\delta Q}{T}$

It has been determined, as stated in the second law of thermodynamics, that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat (δQ) and as work (δW), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal the entropy at the end of the cycle. In the irreversible case, entropy will be created in the system, and more entropy must be extracted than was added $(\Delta S<0)$ in order to return the system to its original state. In the reversible case, no entropy is created and the amount of entropy added is equal to the amount extracted $(\Delta S=0)$.

If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality.

Proof

The temperature that enters in the denominator of the integrand in the Clausius Inequality is actually the temperature of the external reservoir with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir.

Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoir, the net change in entropy of the "universe", so to say,is, $dS_{Total}=dS_{Sys} +dS_{Res} \geq 0$.

When the system takes in heat by an infinitesimal amount $\delta Q_{1}$($\geq 0$), for the net change in entropy $dS_{Total_{1}}$ in this step to be positive, the temperature of the "hot" reservoir $T_{Hot}$ needs to be slightly greater than the temperature of the system at that instant. If the temperature of the system is given by $T_{1}$ at that instant, then $dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}$, and $T_{Hot}\geq T_{1}$ forces us to have:

$-dS_{Res_{1}} =\frac{\delta Q_{1}}{T_{Hot}}\leq \frac{\delta Q_{1}}{T_{1}} = dS_{Sys_{1}}$

This means the magnitude of the entropy "loss" from the reservoir, $|dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}}$ is less than the magnitude of the entropy gain $dS_{Sys_{1}}$($\geq 0$) by the system:

Similarly, when the system at temperature $T_{2}$ expels heat in magnitude $-\delta Q_{2}$ ($\delta Q_{2}\leq 0$) into a colder reservoir (at temperature $T_{Cold}\leq T_{2}$) in an infinitesimal step, then again, for the Second Law of Thermodynamics to hold,we would have, in an exactly similar manner:

$-dS_{Res_{2}}=\frac{\delta Q_{2}}{T_{Cold}}\leq \frac{\delta Q_{2}}{T_{2}}= dS_{Sys_{2}}$

Here,the amount of heat 'absorbed' by the system is given by $\delta Q_{2}$($\leq 0$), signifying that heat is transferring from the system to the reservoir, with $dS_{Sys_{2}}\leq 0$. The magnitude of the entropy gained by the reservoir, $dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}$ is greater than the magnitude of the entropy loss of the system $|dS_{Sys_{2}}|$

Since the total change in entropy for the system is 0 in a cyclic process, if we add all the infinitesimal steps of heat intake and heat expulsion from the reservoir, signified by the previous two equations, with the temperature of the reservoir at each instant given by $T$, we would have,

$-\oint dS_{Res}= \oint \frac{\delta Q}{T}\leq \oint dS_{Sys}=0$

In particular.

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

Hence, we prove the Clausius Theorem.

We summarize the following, (the inequality in the third statement below, being obviously guaranteed by the second law of thermodynamics,which is the basis of our calculation) ,

$\oint dS_{Res}\geq 0$
$\oint dS_{Sys}=0$ (as hypothesised)
$\oint dS_{Total}=\oint dS_{Res}+\oint dS_{Sys}\geq 0$

For a reversible cyclic process, there is no generation of entropy in each of the infinitesimal heat transfer processes, and thus we would have the equality

$\oint \frac{\delta Q_{rev}}{T}=0$

Thus, the Clausius Inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.