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A reversible process is one carried out in infinitesimal steps after which, when undone, both the system and surroundings (that is, the world) remain unchanged (see the example of gas expansion-compression below). Although true reversible change cannot be realized in practice, it can always be approximated. As a process is carried out in a more reversible manner, the value of w approaches its maximum possible value, and q approaches its minimum possible value. Although q is not a state function, the quotient qrev/T is, and is known as the entropy.
A change is said to occur reversibly when it can be carried out in a series of infinitesimal steps, each one of which can be undone by making a similarly minute change to the conditions that bring the change about. For example, the reversible expansion of a gas can be achieved by reducing the external pressure in a series of infinitesimal steps; reversing any step will restore the system and the surroundings to their previous state. Similarly, heat can be transferred reversibly between two bodies by changing the temperature difference between them in infinitesimal steps each of which can be undone by reversing the temperature difference.
The most widely cited example of an irreversible change is the free expansion of a gas into a vacuum. Although the system can always be restored to its original state by recompressing the gas, this would require that the surroundings perform work on the gas. Since the gas does no work on the surrounding in a free expansion (the external pressure is zero, so (PΔV = 0),) there will be a permanent change in the surroundings. Another example of irreversible change is the conversion of mechanical work into frictional heat; there is no way, by reversing the motion of a weight along a surface, that the heat released due to friction can be restored to the system.
: Reversible vs. Irreversible Expansions and Compressions. (CC BY-NC;Ümit Kaya)</p>
<p>These diagrams show the same expansion and compression ±ΔV carried out in different numbers of steps ranging from a single step at the top to an “infinite” number of steps at the bottom. As the number of steps increases, the processes become less irreversible; that is, the difference between the work done in expansion and that required to re-compress the gas diminishes. In the limit of an ”infinite” number of steps (bottom), these work terms are identical, and both the system and surroundings (the “world”) are unchanged by the expansion-compression cycle. In all other cases the system (the gas) is restored to its initial state, <strong>but the <em>surroundings</em> are forever changed.</strong></p>
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<p>Definition: Reversible Changes</p>
<p>A reversible change is one carried out in such as way that, when undone, both the system and surroundings (that is, the world) remain unchanged.</p>
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<p>It should go without saying, of course, that any process that proceeds in infinitesimal steps would take infinitely long to occur, so thermodynamic reversibility is an idealization that is never achieved in real processes, except when the system is already at equilibrium, in which case no change will occur anyway! So why is the concept of a reversible process so important?</p>
<p>The answer can be seen by recalling that the change in the internal energy that characterizes any process can be distributed in an infinity of ways between heat flow across the boundaries of the system and work done on or by the system, as expressed by the First Law of thermodynamics</p>
<p><ΔU = q + w. label{first} ></p>
<p>Each combination of <em>q</em> and <em>w</em> represents a different pathway between the initial and final states. It can be shown that as a process such as the expansion of a gas is carried out in successively longer series of smaller steps, the absolute value of <em>q</em> approaches a minimum, and that of <em>w</em> approaches a maximum that is characteristic of the particular process. Thus when a process is carried out reversibly, the <em>w</em>-term in Equation<br />
ef{first} has its greatest possible value, and the <em>q</em>-term is at its smallest. These special quantities (w_{max}) and (q_{min}) (which we denote as (q_{rev}) and pronounce “q-reversible”) have unique values for any given process and are therefore state functions.</p>
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<h2>Work and Reversibility</h2>
<p>Changes in entropy ((ΔS)), together with changes in enthalpy ((ΔH)), enable us to predict in which direction a jonathanlewisforcongress.comical or physical change will occur spontaneously. Before discussing how to do so, however, we must understand the difference between a reversible process and an irreversible one. In a reversible process, every intermediate state between the extremes is an equilibrium state, regardless of the direction of the change. In contrast, an irreversible process is one in which the intermediate states are not equilibrium states, so change occurs spontaneously in only one direction. As a result, a reversible process can change direction at any time, whereas an irreversible process cannot. When a gas expands reversibly against an external pressure such as a piston, for example, the expansion can be reversed at any time by reversing the motion of the piston; once the gas is compressed, it can be allowed to expand again, and the process can continue indefinitely. In contrast, the expansion of a gas into a vacuum ((P_{ext} = 0)) is irreversible because the external pressure is measurably less than the internal pressure of the gas. No equilibrium states exist, and the gas expands irreversibly. When gas escapes from a microscopic hole in a balloon into a vacuum, for example, the process is irreversible; the direction of airflow cannot change.</p>
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Because work done during the expansion of a gas depends on the opposing external pressure ((w = P_{ext}ΔV)), work done in a reversible process is always equal to or greater than work done in a corresponding irreversible process: (w_{rev} ≥ w_{irrev}). Whether a process is reversible or irreversible, the first law of thermodynamics holds (Equation
ef{first}). Because (U) is a state function, the magnitude of (ΔU) does not depend on reversibility and is independent of the path taken. So
<egin{align} ΔU &= q_{rev} + w_{rev} \<4pt> &= q_{irrev} + w_{irrev} label{Eq1} end{align}>
Work done in a reversible process is always equal to or greater than work done in a corresponding irreversible process:
In other words, (ΔU) for a process is the same whether that process is carried out in a reversible manner or an irreversible one. We now return to our earlier definition of entropy, using the magnitude of the heat flow for a reversible process ((q_{rev})) to define entropy quantitatively.
: Note that the reversible condition implies (w_{max}) and(q_{min}). The impossibility of extracting all of the internal energy as work is essentially a statement of the Second Law.(CC BY-NC;Ümit Kaya)</p>
<p>For a process that reversibly exchanges a quantity of heat (q_{rev}) with the surroundings, the entropy change is defined as</p>
<p>< color{red} Delta S = dfrac{q_{rev}}{T} label{23.2.1}></p>
<p>This is the basic way of evaluating (ΔS) for <em>constant-temperature processes</em> such as phase changes, or the isothermal expansion of a gas. For processes in which the temperature is not constant such as heating or cooling of a substance, the equation must be integrated over the required temperature range, as discussed below.</p>
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<p>Q&A: If no real process can take place reversibly, what use is an expression involving (q_{rev})?</p>
<p>This is a rather fine point that you should understand: although transfer of heat between the system and surroundings is impossible to achieve in a truly reversible manner, this idealized pathway is only crucial for the definition of (ΔS); by virtue of its being a state function, the same value of (ΔS) will apply when the system undergoes the same net change via any pathway. For example, the entropy change a gas undergoes when its volume is doubled at constant temperature will be the same regardless of whether the expansion is carried out in 1000 tiny steps (as reversible as patience is likely to allow) or by a single-step (as irreversible a pathway as you can get!) expansion into a vacuum (Figure (PageIndex{1})).</p>
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<p>Entropy and “disorder”</p>
<p>Entropy is still described, particularly in older textbooks, as a measure of disorder. In a narrow technical sense this is correct, since the spreading and sharing of thermal energy does have the effect of randomizing the disposition of thermal energy within a system. But to simply equate entropy with “disorder” without further qualification is extremely misleading because it is far too easy to forget that entropy (and thermodynamics in general) applies only to molecular-level systems capable of exchanging thermal energy with the surroundings. Carrying these concepts over to macro systems may yield compelling analogies, but it is no longer science. It is far better to avoid the term “disorder” altogether in discussing entropy.</p>
<p><figure class=)
 label{23.2.4}></p>
<h2>The Relationship between Internal Energy and Entropy</h2>
<p>Because the quantity of heat transferred ((q_{rev})) is directly proportional to the absolute temperature of an object ((T)) ((q_{rev} propto T)), the hotter the object, the greater the amount of heat transferred. Moreover, adding heat to a system increases the kinetic energy of the component atoms and molecules and hence their disorder ((ΔS ∝ q_{rev})). Combining these relationships for any reversible process,</p>
<p><q_{ extrm{rev}}=TDelta S></p>
<p>and</p>
<p><Delta S=dfrac{q_{ extrm{rev}}}{T} label{Eq2}></p>
<p>Because the numerator (qrev) is expressed in units of energy (joules), the units of ΔS are joules/kelvin (J/K). Recognizing that the work done in a reversible process at constant pressure is</p>
<p><w_{rev} = −PΔV,></p>
<p>we can express Equation (<br />
ef{Eq1}) as follows:</p>
<p><egin{align} ΔU &= q_{rev} + w_{rev} \<4pt> &= TΔS − PΔV label{Eq3} end{align} ></p>
<p>Thus the change in the internal energy of the system is related to the change in entropy, the absolute temperature, and the (PV) work done.</p>
<p>To illustrate the use of Equation (<br />
ef{Eq2}) and Equation (<br />
ef{Eq3}), we consider two reversible processes before turning to an irreversible process. When a sample of an ideal gas is allowed to expand reversibly at constant temperature, heat must be added to the gas during expansion to keep its T constant (Figure (PageIndex{5})). The internal energy of the gas does not change because the temperature of the gas does not change; that is, ΔU = 0 and qrev = −wrev. During expansion, ΔV > 0, so the gas performs work on its surroundings:</p>
<p><w_{rev} = −PΔV
<figure class=)
: Expansion of Gas at Constant Temperature. (CC BY-SA-NC; Anonymous by request)</p>
<p>In the initial state (top), the temperatures of a gas and the surroundings are the same. During the reversible expansion of the gas, heat must be added to the gas to maintain a constant temperature. Thus the internal energy of the gas does not change, but work is performed on the surroundings. In the final state (bottom), the temperature of the surroundings is lower because the gas has absorbed heat from the surroundings during expansion.</p>
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Now consider the reversible melting of a sample of ice at 0°C and 1 atm. The enthalpy of fusion of ice is 6.01 kJ/mol, which means that 6.01 kJ of heat are absorbed reversibly from the surroundings when 1 mol of ice melts at 0°C, as illustrated in Figure (PageIndex{6}). The surroundings constitute a sample of low-density carbon foam that is thermally conductive, and the system is the ice cube that has been placed on it. The direction of heat flow along the resulting temperature gradient is indicated with an arrow. From Equation (
ef{Eq2}), we see that the entropy of fusion of ice can be written as follows:
