How Does The Entropy Of The System Change When A Solid Melts, A Gas Liquefies, A Solid Sublimes?

Chapter 16. Thermodynamics

sixteen.2 Entropy

Learning Objectives

By the cease of this section, y'all will be able to:

Define entropy
Explain the relationship between entropy and the number of microstates
Predict the sign of the entropy change for chemical and concrete processes

In 1824, at the age of 28, Nicolas Léonard Sadi Carnot (Figure 1) published the results of an extensive study regarding the efficiency of steam heat engines. In a later review of Carnot'south findings, Rudolf Clausius introduced a new thermodynamic belongings that relates the spontaneous heat flow accompanying a process to the temperature at which the procedure takes place. This new holding was expressed as the ratio of the reversible heat (q _rev) and the kelvin temperature (T). The term reversible process refers to a process that takes place at such a ho-hum charge per unit that it is always at equilibrium and its direction tin be changed (information technology can be "reversed") by an infinitesimally small modify is some condition. Notation that the idea of a reversible process is a ceremonial required to support the development of various thermodynamic concepts; no existent processes are truly reversible, rather they are classified every bit irreversible.

A portrait of Rudolf Clasius is shown. — **Figure 1.** (a) Nicholas Léonard Sadi Carnot's enquiry into steam-powered machinery and (b) Rudolf Clausius's after study of those findings led to groundbreaking discoveries about spontaneous heat flow processes.

Similar to other thermodynamic backdrop, this new quantity is a state part, and and then its modify depends but upon the initial and final states of a system. In 1865, Clausius named this holding entropy (Due south) and divers its change for whatever process as the following:

[latex]{\Delta}S = \frac{q_{\text{rev}}}{T}[/latex]

The entropy change for a existent, irreversible process is so equal to that for the theoretical reversible process that involves the same initial and final states.

Entropy and Microstates

Following the work of Carnot and Clausius, Ludwig Boltzmann developed a molecular-calibration statistical model that related the entropy of a system to the number of microstates possible for the organisation. A microstate (West) is a specific configuration of the locations and energies of the atoms or molecules that comprise a system like the following:

[latex]S = 1000\;\text{ln}\;W[/latex]

Here k is the Boltzmann abiding and has a value of 1.38 × 10⁻²³ J/Grand.

As for other land functions, the modify in entropy for a process is the departure betwixt its terminal (South _f) and initial (S _i) values:

[latex]{\Delta}South = S_{\text{f}}\;-\;S_{\text{i}} = thou\;\text{ln}\;W_{\text{f}}\;-\;k\;\text{ln}\;W_{\text{i}} = yard\;\text{ln}\;\frac{W_{\text{f}}}{W_{\text{i}}}[/latex]

For processes involving an increment in the number of microstates, Due west _f > W _i, the entropy of the organisation increases, ΔS > 0. Conversely, processes that reduce the number of microstates, W _f < W _i, yield a decrease in system entropy, ΔSouth < 0. This molecular-calibration interpretation of entropy provides a link to the probability that a process volition occur as illustrated in the next paragraphs.

Consider the general case of a system comprised of Northward particles distributed among n boxes. The number of microstates possible for such a system is n^Northward . For example, distributing four particles amidst two boxes will result in 2⁴ = 16 different microstates as illustrated in Figure ii. Microstates with equivalent particle arrangements (not considering private particle identities) are grouped together and are called distributions. The probability that a organisation volition be with its components in a given distribution is proportional to the number of microstates within the distribution. Since entropy increases logarithmically with the number of microstates, the nigh probable distribution is therefore the one of greatest entropy.

Five rows of diagrams that look like dominoes are shown and labeled a, b, c, d, and e. Row a has one — **Figure two.** The sixteen microstates associated with placing iv particles in two boxes are shown. The microstates are nerveless into five distributions—(a), (b), (c), (d), and (e)—based on the numbers of particles in each box.

For this system, the well-nigh likely configuration is one of the six microstates associated with distribution (c) where the particles are evenly distributed betwixt the boxes, that is, a configuration of 2 particles in each box. The probability of finding the organization in this configuration is or [latex]\frac{6}{16}[/latex] or [latex]\frac{3}{viii}[/latex]. The least probable configuration of the system is one in which all four particles are in 1 box, corresponding to distributions (a) and (d), each with a probability of [latex]\frac{1}{16}[/latex]. The probability of finding all particles in only one box (either the left box or right box) is then [latex](\frac{1}{16}\;+\;\frac{i}{16}) = \frac{2}{xvi}[/latex] or [latex]\frac{1}{8}[/latex].

As yous add more than particles to the arrangement, the number of possible microstates increases exponentially (2^N). A macroscopic (laboratory-sized) organisation would typically consist of moles of particles (North ~ 10²³), and the corresponding number of microstates would exist staggeringly huge. Regardless of the number of particles in the arrangement, yet, the distributions in which roughly equal numbers of particles are constitute in each box are always the most probable configurations.

The previous description of an ideal gas expanding into a vacuum (Chapter 16.1 Spontaneity) is a macroscopic example of this particle-in-a-box model. For this system, the nearly probable distribution is confirmed to be the one in which the matter is most uniformly dispersed or distributed betwixt the two flasks. The spontaneous process whereby the gas contained initially in one flask expands to fill both flasks equally therefore yields an increase in entropy for the system.

A similar arroyo may be used to depict the spontaneous flow of heat. Consider a system consisting of two objects, each containing two particles, and two units of energy (represented every bit "*") in Figure 3. The hot object is comprised of particles A and B and initially contains both free energy units. The cold object is comprised of particles C and D, which initially has no energy units. Distribution (a) shows the three microstates possible for the initial state of the system, with both units of energy contained within the hot object. If one of the two free energy units is transferred, the effect is distribution (b) consisting of 4 microstates. If both energy units are transferred, the result is distribution (c) consisting of iii microstates. And so, we may describe this organisation by a total of ten microstates. The probability that the estrus does not flow when the two objects are brought into contact, that is, that the system remains in distribution (a), is [latex]\frac{3}{ten}[/latex]. More probable is the menstruation of oestrus to yield i of the other ii distribution, the combined probability being [latex]\frac{vii}{10}[/latex]. The almost likely result is the menstruum of oestrus to yield the uniform dispersal of free energy represented by distribution (b), the probability of this configuration being [latex]\frac{iv}{x}[/latex]. As for the previous example of thing dispersal, extrapolating this handling to macroscopic collections of particles dramatically increases the probability of the uniform distribution relative to the other distributions. This supports the mutual observation that placing hot and common cold objects in contact results in spontaneous heat catamenia that ultimately equalizes the objects' temperatures. And, once more, this spontaneous process is as well characterized by an increment in system entropy.

Three rows labeled a, b, and c are shown and each contains rectangles with two sides where the left side is labeled, — **Figure three.** This shows a microstate model describing the catamenia of rut from a hot object to a cold object. (a) Before the estrus flow occurs, the object comprised of particles A and B contains both units of energy and every bit represented by a distribution of three microstates. (b) If the heat flow results in an fifty-fifty dispersal of energy (i energy unit transferred), a distribution of four microstates results. (c) If both energy units are transferred, the resulting distribution has three microstates.

Example 1

Conclusion of ΔSouthward
Consider the system shown here. What is the change in entropy for a process that converts the organization from distribution (a) to (c)?

Solution
We are interested in the post-obit change:

The initial number of microstates is one, the final six:

[latex]{\Delta}S = 1000\;\text{ln}\;\frac{W_{\text{c}}}{W_{\text{a}}} = ane.38\;\times\;10^{-23}\;\text{J}/\text{K}\;\times\;\text{ln}\;\frac{half dozen}{1} = 2.47\;\times\;10^{-23}\;\text{J}/\text{K}[/latex]

The sign of this result is consequent with expectation; since there are more than microstates possible for the concluding state than for the initial state, the change in entropy should be positive.

Check Your Learning
Consider the system shown in Figure three. What is the modify in entropy for the process where all the energy is transferred from the hot object (AB) to the common cold object (CD)?

Predicting the Sign of ΔS

The relationships between entropy, microstates, and matter/free energy dispersal described previously allow us to make generalizations regarding the relative entropies of substances and to predict the sign of entropy changes for chemical and physical processes. Consider the phase changes illustrated in Figure iv. In the solid phase, the atoms or molecules are restricted to well-nigh stock-still positions with respect to each other and are capable of only pocket-sized oscillations well-nigh these positions. With essentially fixed locations for the system's component particles, the number of microstates is relatively small. In the liquid phase, the atoms or molecules are free to move over and around each other, though they remain in relatively close proximity to one another. This increased liberty of motility results in a greater variation in possible particle locations, then the number of microstates is correspondingly greater than for the solid. Every bit a result, S _liquid > Due south _solid and the process of converting a substance from solid to liquid (melting) is characterized by an increase in entropy, ΔS > 0. By the aforementioned logic, the reciprocal process (freezing) exhibits a subtract in entropy, ΔS < 0.

Three stoppered flasks are shown with right and left-facing arrows in between each; the first is labeled above as, — **Effigy 4.** The entropy of a substance increases (Δ*Southward* > 0) equally it transforms from a relatively ordered solid, to a less-ordered liquid, and then to a still less-ordered gas. The entropy decreases (Δ*Due south* < 0) equally the substance transforms from a gas to a liquid and and then to a solid.

At present consider the vapor or gas stage. The atoms or molecules occupy a much greater volume than in the liquid phase; therefore each atom or molecule can be found in many more locations than in the liquid (or solid) stage. Consequently, for any substance, South _gas > S _liquid > S _solid, and the processes of vaporization and sublimation likewise involve increases in entropy, ΔDue south > 0. Likewise, the reciprocal phase transitions, condensation and deposition, involve decreases in entropy, ΔS < 0.

According to kinetic-molecular theory, the temperature of a substance is proportional to the average kinetic energy of its particles. Raising the temperature of a substance will issue in more extensive vibrations of the particles in solids and more rapid translations of the particles in liquids and gases. At higher temperatures, the distribution of kinetic energies among the atoms or molecules of the substance is too broader (more dispersed) than at lower temperatures. Thus, the entropy for whatsoever substance increases with temperature (Figure v).

Two graphs are shown. The y-axis of the left graph is labeled, — **Figure five.** Entropy increases every bit the temperature of a substance is raised, which corresponds to the greater spread of kinetic energies. When a substance melts or vaporizes, information technology experiences a significant increase in entropy.

Endeavor this simulator with interactive visualization of the dependence of particle location and freedom of motion on physical state and temperature.

The entropy of a substance is influenced by structure of the particles (atoms or molecules) that comprise the substance. With regard to atomic substances, heavier atoms possess greater entropy at a given temperature than lighter atoms, which is a effect of the relation between a particle'due south mass and the spacing of quantized translational energy levels (which is a topic beyond the scope of our treatment). For molecules, greater numbers of atoms (regardless of their masses) increase the means in which the molecules can vibrate and thus the number of possible microstates and the organization entropy.

Finally, variations in the types of particles affects the entropy of a organisation. Compared to a pure substance, in which all particles are identical, the entropy of a mixture of two or more unlike particle types is greater. This is considering of the additional orientations and interactions that are possible in a system comprised of nonidentical components. For case, when a solid dissolves in a liquid, the particles of the solid experience both a greater liberty of motion and additional interactions with the solvent particles. This corresponds to a more uniform dispersal of matter and energy and a greater number of microstates. The procedure of dissolution therefore involves an increase in entropy, ΔS > 0.

Because the various factors that affect entropy allows us to make informed predictions of the sign of ΔS for diverse chemical and physical processes as illustrated in Example 2.

Example ii

Predicting the Sign of ∆Due south
Predict the sign of the entropy change for the following processes. Indicate the reason for each of your predictions.

(a) Ane mole liquid water at room temperature ⟶⟶ one mole liquid water at 50 °C

(b) [latex]\text{Ag}^{+}(aq)\;+\;\text{Cl}^{-}(aq)\;{\longrightarrow}\;\text{AgCl}(s)[/latex]

(c) [latex]\text{C}_6\text{H}_6(50)\;+\;\frac{15}{2}\text{O}_2(g)\;{\longrightarrow}\;half-dozen\text{CO}_2(thou)\;+\;iii\text{H}_2\text{O}(l)[/latex]

(d) [latex]\text{NH}_3(south)\;{\longrightarrow}\;\text{NH}_3(l)[/latex]

Solution
(a) positive, temperature increases

(b) negative, reduction in the number of ions (particles) in solution, decreased dispersal of matter

(d) positive, phase transition from solid to liquid, net increase in dispersal of matter

Bank check Your Learning
Predict the sign of the enthalpy change for the post-obit processes. Give a reason for your prediction.

(a) [latex]\text{NaNO}_3(south)\;{\longrightarrow}\;\text{Na}^{+}(aq)\;+\;\text{NO}_3^{\;\;-}(aq)[/latex]

(b) the freezing of liquid water

(d) [latex]\text{CaCO}(s)\;{\longrightarrow}\;\text{CaO}(due south)\;+\;\text{CO}_2(g)[/latex]

Respond:

(a) Positive; The solid dissolves to requite an increment of mobile ions in solution. (b) Negative; The liquid becomes a more ordered solid. (c) Positive; The relatively ordered solid becomes a gas. (d) Positive; At that place is a cyberspace production of one mole of gas.

Key Concepts and Summary

Entropy (South) is a state function that tin exist related to the number of microstates for a system (the number of ways the system tin be arranged) and to the ratio of reversible estrus to kelvin temperature. It may be interpreted equally a mensurate of the dispersal or distribution of matter and/or energy in a system, and it is oftentimes described as representing the "disorder" of the system.

For a given substance, S _solid < S _liquid < S _gas in a given physical state at a given temperature, entropy is typically greater for heavier atoms or more circuitous molecules. Entropy increases when a arrangement is heated and when solutions class. Using these guidelines, the sign of entropy changes for some chemical reactions may be reliably predicted.

Key Equations

[latex]{\Delta}S = \frac{q_{\text{rev}}}{T}[/latex]
[latex]S = k\;\text{ln}\;West[/latex]
[latex]{\Delta}Southward = k\;\text{ln}\;\frac{W_{\text{f}}}{W_{\text{i}}}[/latex]

Chemistry End of Chapter Exercises

In Figure ii all possible distributions and microstates are shown for four different particles shared betwixt 2 boxes. Determine the entropy change, ΔDue south, if the particles are initially evenly distributed betwixt the two boxes, but upon redistribution all end up in Box (b).
In Effigy ii all of the possible distributions and microstates are shown for four dissimilar particles shared between ii boxes. Determine the entropy change, ΔSouthward, for the system when it is converted from distribution (b) to distribution (d).
How does the process described in the previous item relate to the system shown in Chapter sixteen.one Spontaneity?
Consider a organization similar to the 1 in Figure 2, except that it contains six particles instead of four. What is the probability of having all the particles in merely one of the ii boxes in the case? Compare this with the similar probability for the organization of iv particles that we accept derived to be equal to [latex]\frac{1}{8}[/latex]. What does this comparison tell us nigh fifty-fifty larger systems?
Consider the system shown in Figure three. What is the change in entropy for the process where the energy is initially associated just with particle A, but in the final state the free energy is distributed between two different particles?
Consider the organisation shown in Figure 3. What is the modify in entropy for the procedure where the energy is initially associated with particles A and B, and the free energy is distributed between two particles in different boxes (one in A-B, the other in C-D)?
Arrange the following sets of systems in social club of increasing entropy. Assume one mole of each substance and the same temperature for each member of a set.
(a) H₂(m), HBrO_four(g), HBr(g)

(b) H_twoO(l), H₂O(g), H₂O(s)

(c) He(g), Cl₂(k), P₄(thou)
At room temperature, the entropy of the halogens increases from I_two to Br₂ to Cl₂. Explain.
Consider two processes: sublimation of I_ii(s) and melting of I₂(s) (Notation: the latter process can occur at the same temperature but somewhat higher pressure level).
[latex]\text{I}_2(s)\;{\longrightarrow}\;\text{I}_2(grand)[/latex]

[latex]\text{I}_2(s)\;{\longrightarrow}\;\text{I}_2(l)[/latex]

Is ΔS positive or negative in these processes? In which of the processes volition the magnitude of the entropy change exist greater?
Indicate which substance in the given pairs has the higher entropy value. Explicate your choices.
(a) C₂H_fiveOH(l) or C_threeH₇OH(l)

(b) C_twoH_vOH(50) or C₂H₅OH(g)

(c) 2H(m) or H(g)
Predict the sign of the entropy change for the following processes.
(a) An ice cube is warmed to near its melting point.

(b) Exhaled breath forms fog on a cold morn.

(c) Snow melts.
Predict the sign of the entropy alter for the following processes. Give a reason for your prediction.
(a) [latex]\text{Pb}^{2+}(aq)\;+\;\text{South}^{2-}(aq)\;{\longrightarrow}\;\text{PbS}(s)[/latex]

(b) [latex]ii\text{Fe}(s)\;+\;three\text{O}_2(g)\;{\longrightarrow}\;\text{Fe}_2\text{O}_3(s)[/latex]

(c) [latex]2\text{C}_6\text{H}_{fourteen}(l)\;+\;19\text{O}_2(g)\;{\longrightarrow}\;14\text{H}_2\text{O}(grand)\;+\;12\text{CO}_2(g)[/latex]
Write the balanced chemic equation for the combustion of methane, CH₄(grand), to requite carbon dioxide and water vapor. Explicate why it is difficult to predict whether ΔDue south is positive or negative for this chemic reaction.
Write the balanced chemical equation for the combustion of benzene, C_{half dozen}H₆(fifty), to give carbon dioxide and water vapor. Would y'all expect ΔS to be positive or negative in this process?

Glossary

entropy (S): state function that is a measure of the thing and/or energy dispersal within a system, determined by the number of arrangement microstates ofttimes described equally a measure of the disorder of the system

microstate (W): possible configuration or arrangement of matter and free energy inside a system

reversible process: procedure that takes place so slowly equally to be capable of reversing direction in response to an infinitesimally small change in atmospheric condition; hypothetical construct that can only exist approximated by real processes removed

Solutions

Answers to Chemistry Cease of Chapter Exercises

two. At that place are four initial microstates and four final microstates.
[latex]{\Delta}S = k\;\text{ln}\;\frac{W_{\text{f}}}{W_{\text{i}}} = 1.38\;\times\;x^{-23}\;\text{J}/\text{K}\;\times\;\text{ln}\;\frac{4}{four} = 0[/latex]

iv. The probability for all the particles to be on ane side is [latex]\frac{1}{32}[/latex]. This probability is noticeably lower than the [latex]\frac{1}{8}[/latex] result for the four-particle organisation. The conclusion nosotros can make is that the probability for all the particles to stay in only ane office of the system volition subtract rapidly as the number of particles increases, and, for instance, the probability for all molecules of gas to assemble in only ane side of a room at room temperature and pressure level is negligible since the number of gas molecules in the room is very large.

6. At that place is merely ane initial land. For the final land, the energy can exist contained in pairs A-C, A-D, B-C, or B-D. Thus, there are four final possible states.
[latex]{\Delta}S = k\;\text{ln}\;(\frac{W_{\text{f}}}{W_{\text{i}}}) = 1.38\;\times\;10^{-23}\;\text{J}/\text{K}\;\times\;\text{ln}\;(\frac{four}{one}) = 1.91\;\times\;ten^{-23}\;\text{J}/\text{Grand}[/latex]

viii. The masses of these molecules would suggest the opposite trend in their entropies. The observed trend is a event of the more significant variation of entropy with a physical land. At room temperature, I_two is a solid, Br₂ is a liquid, and Cl₂ is a gas.

x. (a) C_iiiH₇OH(50) as it is a larger molecule (more than complex and more massive), then more microstates describing its motions are available at any given temperature. (b) C₂H₅OH(g) as it is in the gaseous state. (c) 2H(g), since entropy is an extensive property, and and then ii H atoms (or two moles of H atoms) possess twice as much entropy equally one atom (or one mole of atoms).

12. (a) Negative. The relatively ordered solid precipitating decreases the number of mobile ions in solution. (b) Negative. There is a net loss of three moles of gas from reactants to products. (c) Positive. In that location is a net increment of seven moles of gas from reactants to products.

14. [latex]\text{C}_6\text{H}_6(fifty)\;+\;7.5\text{O}_2(1000)\;{\longrightarrow}\;3\text{H}_2\text{O}(g)\;+\;6\text{CO}_2(g)[/latex]
There are 7.five moles of gas initially, and three + six = nine moles of gas in the cease. Therefore, information technology is probable that the entropy increases as a result of this reaction, and ΔDue south is positive.