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==Free energy: the Gibbs function==
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__NOTOC__
{{wiktionarypar|sandbox}}
'''Sandbox''' may refer to:


* For cats: ], an indoor box for cats to relieve themselves
In the <u>previous section</u> we saw that it is the sum of the entropy changes of the system and surroundings that determines whether a process will occur spontaneously. In chemical thermodynamics we prefer to focus our attention on the system rather than the surroundings, and would like to avoid having to calculate the entropy change of the surroundings explicity. The key to doing this is to define a new ''state function'' known as the ''Gibbs free energy''
* For children: ], a wide, shallow playground construction to hold sand for children to play in.
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==Computing==
{|
* ], a virtual container in which untrusted programs can be safely run
| align="center"| ''G'' = ''H'' – ''TS''
* ], an online environment in which code or content changes can be tested without affecting the original system(s)
| (15)
* ], in Google Internet search rankings
|}
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* ], a game level editor for the game ]
* ], a derogatory term for a microelectronics fabrication facility that employs out of date processes, or, especially, shoddy or ad hoc process controls


==Military==
Since ''H'', ''T'' and ''S'' are all state functions, ''G'' is a function of state. This relation is more usefully expressed in terms of the change
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==Music==
<center>Δ''G'' = Δ''H'' – ''T''Δ''S'' (16) Must know this!</center>
*], a 1987 release by the band Guided By Voices
*], a Canadian rock music group, originally from Nova Scotia
*], an American rock music group, originally from Syracuse, New York
*"Sandbox Magician", a song from the 1998 '']'' EP by The Dillinger Escape Plan


{{disambig}}
The Δ-quantities refer to changes of the ''system''; in particular, Δ''S'' is the entropy change of the system and (being a state function) has a fixed value for a given change. Note, however, that the Δ''H'' term represents the heat q<sub>''p''</sub> that is gained or lost by the system, and that this same quantity of heat (but with opposite sign) is lost or gained by the surroundings.
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]
For a process that takes place at a temperature ''T'', we can write the total entropy change as the sum of those for the surroundings and for the system, expressed in terms of the heat that flows into
]

]
<center>] (16)</center>
]

]
Multiplying through by –T , we obtain
]

]
<center>]</center>
]

which expresses the entropy change of the world in terms of thermodynamic properties of the system exclusively. If –TΔS<sub>''total''</sub> is denoted by Δ''G'', then we have Eq. 16 which defines the ''Gibbs free energy'' change for the process.

From the foregoing, you should convince yourself that ''G'' (now often referred to as the ''Gibbs function'' rather than as free energy) will decrease in any process occurring at constant temperature and pressure which is accompanied by an overall increase in the entropy. (The constant temperature and pressure are a consequence of the temperature and the enthalpy appearing in the preceding equation.) Since most chemical and phase changes of interest to chemists take place under such conditions, the Gibbs function is the most useful of all the thermodynamic properties of a substance, and it is closely linked to the equilibrium constant.

{| border="1" cellpadding="5" cellspacing="0"
|''' Problem Example 2'''
|-
|One mole of an ideal gas at 300 K is allowed to expand slowly and isothermally to twice its initial volume. Calculate q, w, ΔS°<sub>''sys''</sub>, ΔS°<sub>''surr''</sub>, and ΔG° for this process.
|-
|-
|'''''Solution:''''' Assume that the process occurs slowly enough that the it can be considered to take place reversibly.
|-
:a) The work done in a reversible expansion is
|
| ]
|-
:so w = (1 mol)(8.314 J mol–1 K–1)(300K)(ln 2) = 1730 J.
|-
:b) In order to maintain a constant temperature, an equivalent quantity of heat must be absorbed by the system: '''q = 1730 J.'''
|-:c) The entropy change of the system is given by <u>Eq. 10 (Part 2)</u>:
|-
|ΔS°<sub>''sys''</sub> = R ln (V<sub>2</sub>/V<sub>1</sub>)
|-
:ΔS°<sub>''sys''</sub> = R ln (V<sub>2</sub>/V<sub>1</sub>) = (8.314 J mol<sup>–1</sup> K<sup>–1</sup>)(ln 2) = '''5.76 J mol<sup>–1</sup> K<sup>–1</sup>'''
|-
:d) <u>As was explained</u> in Part 2, the entropy change of the surroundings, formally defined as q<sub>''rev''</sub>/''T'', is
|-
:ΔS°surr = q/T = (–1730 J) / (300 K) = – '''5.76 J mol<sup>–1</sup> K<sup>–1</sup>.'''
|-
:e) The free energy change is
:ΔG° = ΔH° – T ΔS°<sub>sys</sub> = 0 – (300 K)(5.76 J K<sup>–1</sup>) = –'''1730 J'''
|-
:''Comment'': Recall that for the expansion of an ideal gas, ΔH° = 0. Note also that because the expansion is assumed to occur reversibly, the entropy changes in (c) and (d) cancel out, so that ΔS°<sub>world</sub> = 0.
|}
===Physical meaning of the Gibbs function===

'''Its physical meaning: the maximum work.'''

<u>As we explained</u> in Part 2, the two quantities q<sub>rev</sub> (q<sub>min</sub>) and w<sub>max</sub> associated with reversible processes are state functions. We gave the quotient q<sub>rev</sub>/T a new name, the '''''entropy'''''. What we did not say is that wmax also has its own name, the '''''free energy'''''.

The Gibbs free energy is the maximum useful work (excluding PV work associated with volume changes of the system) that a system can do on the surroundings when the process occurs reversibly at constant temperature and pressure.

This work is done at the expense of the internal energy of the system, and whatever part of that is not extracted as work is exchanged with the surroundings as heat; this latter quantity will have the value –TΔS .

'''The Gibbs function: is it free? Is it energy?'''

The appellation “free energy” for ''G'' has led to so much confusion that many scientists now refer to it simply as the '''''Gibbs function'''''. The “free” part of the name reflects the steam-engine origins of thermodynamics with its interest in converting heat into work: ΔG = w<sub>max</sub>, the maximum amount of energy which can be “freed” from the system to perform useful work. A much more serious difficulty, particularly in the context of chemistry, is that although ''G'' has the ''units'' of energy (joules, or in its intensive form, J mol<sup>–1</sup>), it lacks one of the most important attributes of energy in that it is not ''conserved''. Thus although the free energy always falls when a gas expands or a chemical reaction takes place, there need be no compensating increase in energy anywhere else. Referring to ''G'' as an energy also reinforces the false but widespread notion that a fall in energy must accompany any change. But if we accept that energy is conserved, it is apparent that the only necessary condition for change (whether the dropping of a weight, expansion of a gas, or a chemical reaction) is the ''redistribution of energy''. The quantity –Δ''G'' associated with a process represents the quantity of energy that is “shared and spread”, which as we have already explained is the meaning of the increase in the entropy. The quotient –Δ''G''/''T'' is in fact identical with ΔS<sub>total</sub>, the entropy change of the world, whose increase is the primary criterion for any kind of change.

'''Who was Gibbs, anyway?'''

J. Willard Gibbs is considered the father of modern thermodynamics and the most brilliant American-born scientist of the 19th century. He did most of his work in obscurity, publishing his difficult-to-understand papers in the ''Proceedings of the Connecticut Academy of Sciences'', a little-read journal that was unknown to most of the world.

]

<center>Some Gibbs links: <u>brief bio</u> - <u>scientific biography</u></center>
A remarkably rich account of Gibbs' seemingly gray life and his unusual powers of visualization was written by a noted American poet: Rukeyser, M., ''Willard Gibbs''. Garden City, N.J.: Doubleday Duran and Co., Inc., 1942.


===The standard Gibbs free energy===

In order to make use of free energies to predict chemical changes, we need to know the free energies of the individual components of the reaction. For this purpose we can combine the standard enthalpy of formation and the standard entropy of a substance to get its '''''standard free energy of formation'''''

<center>ΔG<sub>''f''</sub>° =ΔH<sub>''f''</sub>° – TΔS<sub>''f''</sub>°</center>

and then determine the standard Gibbs free energy of the reaction according to

<center>ΔG° = ΔG<sub>''f''(products)</sub>° – ΔG<sub>''f''(reactants)</sub>°</center>
{|align="center"
|As with standard heats of formation, the standard free energy of a substance represents the free energy change associated with the formation of the substance from the elements in their most stable forms as they exist under the standard conditions of 1 atm pressure and 298K.tandard Gibbs free energies of formation are normally found directly from tables. Once the values for all the reactants and products are known, the standard Gibbs free energy change for the reaction is found in the normal way.
|}

The interpretation of ΔG° for a chemical change is very simple. For a reaction A Æ B, one of the following three situations will always apply:

{| border="1" cellpadding="5" cellspacing="0" align="center"
|width=100 align="center" | Δ''G''° < 0
|width=400 |net change to the right: A→B
|-align="Center"
|ΔG° < 0
|net change to the left: A ♦ B
|-align="center"
|ΔG° = 0
|no net change; system is at equilibrium
|}

'''Table 4: Δ''G''° as a criterion for spontaneous change'''
The interpretation of ΔG° for a chemical change is very simple. For a reaction A Æ B, one of the three situations illustrated here will always apply:

Some textbooks and teachers still say that the Δ''G''° and thus the criterion for chemical change depends on the two factors ΔH° and ΔS°, and they sometimes even refer to reactions in which one of these terms dominates as "energy driven" or "entropy driven" processes. '''This can be extremely misleading!''' As explained at the top of this page, '''all processes are entropy driven'''; when –ΔH° exceeds the –TΔS° term, this merely means that the entropy change of the surroundings is the greater contributor to the entropy change of the system.

===Why are chemical reactions affected by the temperature?===

The TΔS° term in the equation ΔG° =ΔH° – TΔS° tells us that the temperature dependence of ΔG° depends almost entirely on the entropy change associated with the process. (We say "almost" because the values of ΔH° and ΔS° are themselves slightly temperature dependent; both gradually increase with temperature). In particular, notice that '''the sign of the entropy change determines whether the reaction becomes more or less spontaneous as the temperature is raised.''' Since the signs of both ΔH° and ΔS° can be positive or negative, we can identify four possibilities as outlined in the table below.

{| border="1" cellpadding="5" cellspacing="2" align="center"
|width = "300" align="center"|'''Exothermic reaction, ΔS > 0'''

]
|width="500" |''' C(graphite) + O<sub>2</sub>(g) CO<sub>2</sub>(g)'''
ΔH° = –393 kJ

ΔS° = +2.9 J K<sup>–1</sup>

ΔG° = –394 kJ at 298 K

This ''combustion reaction'', like most such reactions, is spontaneous at all temperatures. The positive entropy change is due mainly to the greater mass of CO<sub>2</sub> compared to O<sub>2</sub>.
|-style="height:50px"
| align="center"|'''Exothermic reaction, ΔS° < 0'''

]
| '''3 H<sub>2</sub> + N<sub>2</sub> → 2 NH<sub>3</sub>(g)'''

ΔH° = –46.2 kJ

ΔS° = –389 J K–1

ΔG° = –16.4 kJ at 298 K

The decrease in moles of gas in the '''Haber ammonia synthesis''' drives the entropy change negative. Thus higher T, which speeds up the reaction, also reduces its extent.
|-style="height:50px"
|align="center"| '''Endothermic reaction, ΔS° > 0'''

]
|''' N<sub>2</sub>O<sub>4</sub>(g) → 2 NO<sub>2</sub>(g)'''

ΔH° = –57.1 kJ

ΔS° = +176 J K<sup>–1</sup>

ΔG° = +4.8 kJ at 298 K

Dissociation reactions are typically endothermic with positive entropy change. Ultimately, all molecules decompose to their atoms at sufficiently high temperatures.
|-style="height:50px"
|align="center"|'''Endothermic reaction, ΔS° < 0'''

]
|''' 1/2 N<sub>2</sub> + O<sub>2</sub> → NO<sub>2</sub>(g)'''

ΔH° = 33.2 kJ

ΔS° = –249 J K–1

ΔG° = +51.3 kJ at 298 K

Although NO<sub>2</sub> is thermodynamically unstable, the reverse of this reaction is kinetically hindered, so this oxide can exist indefinitely at ordinary temperatures.
|}

{|
''' Table 5: Effects of temperature on reaction spontaniety'''

<center>Must know this!</center>

The plots on the left show how ΔH° and TΔS° combine according to determine the range of temperatures over which a reaction can take place spontaneously. Be sure you understand these graphs and can reproduce them for each of the four sign combinations of ΔH° and ΔS°. These relations are of course governed by the relative entropies of the reactants and products, illustrated schematically by the spacing of the quantized energy states; never forget that it is the ability of thermal energy to spread into as many of these states as possible that ultimately determines the tendency of a process to take place. The column on the right of the table shows an example of each class of reaction.
|}

'''Finding the equilibrium temperature'''

Notice that the reactions which are spontaneous only above or below a certain temperature have identical signs for Δ''H''° and Δ''S''°. In these cases there will be a unique temperature at which ΔH° = TΔS° and thus ΔG° = 0 , corresponding to chemical equilibrium. This temperature is given by

<center>T=ΔH°/ΔS° (19)</center>

and corresponds to the point at which the lines representing ΔH° and TΔS° cross in a plot of these two quantities as a function of the temperature.

{|The values of both ΔH° and ΔS° are themselves somewhat temperature dependent, both increasing with the temperature. This means that substitution of 298 K values of these quantities into Eq. 19 will give only an estimate of the equilibrium temperature if this differs greatly from 298 K.Nevertheless, the general conclusions regarding the temperature dependence of reactions such as those shown in Table 5 are always correct, even if the equilibrium temperature cannot be predicted accurately.
|}

'''Why do most substances have definite melting and boiling points'''

As a crystalline substance is heated it eventually melts to a liquid, and finally vaporizes. The transitions between these phases occur abruptly and at definite temperatures; except at the melting and boiling temperatures, only one of these three phases will be stable— that is, its free energy will be lower than that of either of the other two phases. To understand the reason for this, recall that as the temperature rises, TΔS° increases faster than ΔS°, so the free energy (which of course depends on –TΔS°) always falls with temperature. Furthermore, the temperature dependence of G depends on the entropy (we won’t try to prove this here), so G falls off fastest for the high-entropy gas phase, and for liquids faster than for solids.

''' Fig. 9: Free energy of solid, liquid and gaseous water as a function of temperature.'''
]

These plots of G° as a function of the temperature explain why substances have definite melting and boiling points. The stable phase is always the one that has the lowest Gibbs free energy. The temperatures at which one phase becomes more stable than the others are the melting and boiling points.
In Fig. 9 the free energies of each of the three phases of water are plotted as a function of temperature. At any given temperature, one of these phases will have a lower free energy than either of the others; this will of course be the stable phase at that temperature. The cross-over points where two phases have identical free energies are the temperatures at which both phases can coexist— in other words, the melting and boiling temperatures. At any other temperature, there is only a single stable phase.

===Free energy, concentrations and escaping tendency===

The free energy of a pure liquid or solid at 1 atm pressure is just its molar free energy of formation Δ''G''° multiplied by the number of moles present. For gases and substances in solution, we have to take into account the concentration (which, in the case of gases, is normally expressed in terms of the pressure). We know that the lower the concentration, the greater the entropy (<u>Eq. 12 in Part 2</u>), and thus the smaller the free energy.

'''The free energy of a gas: Standard states'''

The free energy of a gas depends on its pressure; the higher the pressure, the higher the free energy. Thus the free expansion of a gas, a spontaneous process, is accompanied by a fall in the free energy. Using <u>Eq. 11 in Part 2</u> we can express the change in free energy when a gas undergoes a change in pressure from P<sub>1</sub> to P<sub>2</sub> as

<center>] (20)</center>

How can we evaluate the free energy of a specific sample of a gas at some arbitrary pressure? First, recall that the standard molar free energy ''G''° that you would look up in a table refers to a pressure of 1 atm. The free energy per mole of our sample is just the sum of this value and any change in free energy that would occur if the pressure were changed from 1 atm to the pressure of interest

<center>''G''= ''G°''+''RT''ln(''P''<sub>1</sub>/1 atm) (21)</center>

which we normally write in abbreviated form

<center>''G'' = G° + ''RT'' ln ''P'' (22)</center>

'''Escaping tendency'''

The higher the pressure of a gas, the greater will be the tendency of its molecules to leave the confines of the container; we will call this the ''escaping tendency''. Equation (22) above tells us that the pressure of a gas is a directly observable measure of its free energy (G, not G°). Combining these two ideas, we can say that the free energy of a gas is also a measure of its escaping tendency. The latter term is not used in traditional thermdynamics because it is essentially synonymous with the free energy, but it is worth knowing because it helps us appreciate the physical significance of free energy in certain contexts.

'''Thermodynamics of mixing and dilution'''

All substances, given the opportunity to form a homogeneous mixture with other substances, will tend to become more dilute. This can be rationalized simply from elementary statistics; there are more equally probable ways of arranging one hundred black marbles and one hundred white marbles, than two hundred marbles of a single color. For massive objects like marbles this has nothing to do with entropy, of course. But when we are dealing with huge numbers of molecules capable of storing, exchanging and spreading thermal energy, mixing and expansion are definitely entropy-driven processes. It can be argued, in fact, that mixing and expansion are really very similar; after all, when we mix two gases, each is expanding into the space formerly occupied exclusively by the other.

''' Fig. 10: Entropy and free energy of mixing'''

]

The two ideal gases in (c) are separated by a barrier. When the barrier is removed (d), the two gases spontaneously mix. ΔS and ΔG for this process are just twice what they would be for the expansion of a single gas (a) to twice its volume (b).
In terms of the spreading of thermal energy the situation is particularly dramatic; the addition of even a single molecule of B to one mole of a gas A results in a huge increase in the number of energically-identical (degenerate) microstates that correspond to the interchange of every molecule in the gas with the new molecule. When actual molar quantities of two gases mix, the number of new microstates created is beyond comprehension.

]
''' Fig. 11: Energy spreading in expansion and mixing'''

The tendency of a gas to expand is due to the more closely-spaced energy states in the larger volume (b). When one molecule of a different kind is introduced into the gas (c), each microstate in (b) splits into a huge number of energetically-identical new states, denoted (inadequately) by the dashed lines in (c).
The entropy increase that occurs when the concentration of a substance is reduced through dilution or mixing is given by <u>Eq. 12 in Part 2</u>. We can similarly define the '''''Gibbs free energy of dilution or mixing''''' by substituting this equation into the definition of ΔG°:

<center>Δ''G''<sub>''dilution''</sub> = Δ''H''<sub>''dilution''</sub>-''RT''ln(''C''<sub>1</sub>/''C''<sub>2</sub>)(23)</center>

If the substance in question forms an ideal solution with the other components, then ΔHdil is by definition zero, and we can write

<center>Δ''G''<sub>''dilution''</sub> = ''RT''ln(''C''<sub>1</sub>/''C''<sub>2</sub>) (24)</center>

These relations tell us that the dilution of a substance from an initial concentration C<sub>1</sub> to a more dilute concentration C<sub>2</sub> is accompanied by a decrease in the free energy, and thus will occur spontaneously.

''' Fig. 12: The equilibrium state for mixing'''
]

If identical numbers of molecules of two gases are allowed to mix as in c-d of Fig. 10 above, the equilibrium mole fraction of each gas will be 0.5.
By the same token, the spontaneous “un-dilution” of a solution will not occur (we do not expect the tea to diffuse back into the teabag!) However, un-dilution can be forced to occur if some means can be found to supply to the system an amount of energy (in the form of work) equal to ΔG<sub>dil</sub>. An important practical example of this is the metabolic work performed by the kidney in concentrating substances from the blood for excretion in the urine.

To find the free energy of a solute at some arbitrary concentration, we proceed in very much the same way as we did for a gas: we take the sum of the standard free energy, and any change in the free energy that would accompany a change in concentration from the standard state to the actual state of the solution. Using Eq. 24 it is easy to derive an expression analogous to <u>Eq. 22</u>

''G'' = ''G''° + ''RT'' ln ''C'' (25)

which gives the free energy of a solute at some arbitrary concentration ''C'' in terms of its value ''G''° in its standard state.

Although this expression has the same simple form as Eq. 22, its practical application is fraught with difficulties, the major one being that it doesn’t usually give values of ''G'' that are consistent with experiment, especially for solutes that are ionic or are slightly soluble. In such solutions, intermolecular interactions between solute molecules and between solute and solvent bring back the enthalpy term that we left out in deriving Eq. 22 (and thus Eq. 25). In addition, the structural organization of the solution becomes concentration dependent, so that the entropy depends on concentraiton in a more complicated way than is implied by <u>Eq. 12 in Part 2</u>.

'''Activity and standard state of the solute'''

Instead of complicating G° by trying to correct for all of these effects, chemists have chosen to retain its simple form by making a single small change in the form of 25:

<center>''G'' = ''G''° + ''RT'' ln ''a'' (26)</center>

This equation is guaranteed to work, because ''a'', the activity of the solute, is its thermodynamically effective concentration. The relation between the activity and the concentration is given by

<center>''a'' = γC (27)</center>

where γ (gamma) is the activity coefficient. As the solution becomes more dilute, the activity coefficient approaches unity:

<center>] (28)</center>

The price we pay for the simplicity of 26 is that the relation between the concentration and the activity at higher concentrations can be quite complicated, and must be determined experimentally for every different solution.

{|align="center"|The question of what standard state we choose for the solute (that is, at what concentration is G° defined, and in what units is it expressed?) is one that you will wish you had never asked! We might be tempted to use a concentration of 1 molar, but a solution this concentrated would be subject to all kinds of intermolecular interaction effects, and would not make a very practical standard state. These effects could be eliminated by going to the opposite extreme of an “infinitely dilute” solution, but by Eq. 25 this would imply a free energy of minus infinity for the solute, which would be awkward. Chemists have therefore agreed to define the '''''standard state of a solute''''' as one in which the concentration is 1 molar, but all solute-solute interactions are magically switched off, so that g is effectively unity. Since this is physically impossible to achieve, no solution corresponding to this standard state can actually exist, but this turns out to be only a small drawback, and seems to be the best compromise between convenience, utility, and reality.
|}

===Summary: the key concepts developed on this page===

(You are expected to be able to define and explain the significance of terms identified in '''''green sans-serif type'''''.)

1. The '''''Gibbs free energy''''' is a '''''state function''''' defined as G = H – TS .

2. The Gibbs free energy is the '''''maximum useful work''''' that a system can do on the surroundings when the process occurs reversibly at constant temperature and pressure.

3. The practical utility of the Gibbs function is that ΔG for any process is negative if it leads to an increase in the '''''entropy of the world'''''. Thus '''''spontaneous change''''' at a given temperature and pressure can only occur when it would lead to a decrease in ''G''.

4. The sign of the standard free energy change ΔG° of a chemical reaction determines whether the reaction will tend to proceed in the forward or reverse direction.

5. Similarly, the relative signs of ΔG° and ΔS° determine whether the spontaniety of a chemical reaction will be affected by the temperature, and if so, in what way.

6. The existence of sharp melting and boiling points reflects the differing temperature dependancies of the free energies of the solid, liquid, and vapor phases of a pure substance, which are in turn reflect their differing entropies.

Revision as of 17:40, 9 September 2007

For the Misplaced Pages feature, where anyone can experiment with editing Misplaced Pages pages, see Misplaced Pages:Sandbox.

Sandbox may refer to:

  • For cats: Litter box, an indoor box for cats to relieve themselves
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  • In therapy: Sandbox Play is a tool used by child psychologists.
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  • On stage: The Sandbox (play), a 1960 one-act play by Edward Albee

Computing

Military

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