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Gibbs free energy and spontaneity

How the second law of thermodynamics helps us determine whether a process will be spontaneous, and using changes in Gibbs free energy to predict whether a reaction will be spontaneous in the forward or reverse direction (or whether it is at equilibrium!). 

Key points

  • The second law of thermodynamics says that the entropy of the universe always increases for a spontaneous process: delta, start text, S, end text, start subscript, start text, u, n, i, v, e, r, s, e, end text, end subscript, equals, delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, plus, delta, start text, S, end text, start subscript, start text, s, u, r, r, o, u, n, d, i, n, g, s, end text, end subscript, is greater than, 0
  • At constant temperature and pressure, the change in Gibbs free energy is defined as delta, start text, G, end text, equals, delta, start text, H, end text, minus, start text, T, end text, delta, start text, S, end text.
  • When delta, start text, G, end text is negative, a process will proceed spontaneously and is referred to as exergonic.
  • The spontaneity of a process can depend on the temperature.

Spontaneous processes

In chemistry, a spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of a diamond turning into graphite, which can be written as the following reaction:
start text, C, end text, left parenthesis, s, comma, start text, d, i, a, m, o, n, d, end text, right parenthesis, right arrow, start text, C, end text, left parenthesis, s, comma, start text, g, r, a, p, h, i, t, e, end text, right parenthesis
On left, multiple shiny cut diamonds. On right, chunk of black graphitic carbon.
Ever heard the saying, "graphite is forever"? If we waited long enough, we would observe a diamond spontaneously turn into the more stable form of carbon, graphite. Picture from Wikipedia, CC BY-SA 3.0
This reaction takes so long that it is not detectable on the timescale of (ordinary) humans, hence the saying, "diamonds are forever." If we could wait long enough, we should be able to see carbon in the diamond form turn into the more stable but less shiny, graphite form.
Another thing to remember is that spontaneous processes can be exothermic or endothermic. That is another way of saying that spontaneity is not necessarily related to the enthalpy change of a process, delta, start text, H, end text.
How do we know if a process will occur spontaneously? The short but slightly complicated answer is that we can use the second law of thermodynamics. According to the second law of thermodynamics, any spontaneous process must increase the entropy in the universe. This can be expressed mathematically as follows:
delta, start text, S, end text, start subscript, start text, u, n, i, v, e, r, s, e, end text, end subscript, equals, delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, plus, delta, start text, S, end text, start subscript, start text, s, u, r, r, o, u, n, d, i, n, g, s, end text, end subscript, is greater than, 0, space, space, space, space, space, space, space, space, start text, F, o, r, space, a, space, s, p, o, n, t, a, n, e, o, u, s, space, p, r, o, c, e, s, s, end text
Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be a chemical reaction in a beaker. Do we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?)
Luckily, chemists can get around having to determine the entropy change of the universe by defining and using a new thermodynamic quantity called Gibbs free energy.

Gibbs free energy and spontaneity

When a process occurs at constant temperature start text, T, end text and pressure start text, P, end text, we can rearrange the second law of thermodynamics and define a new quantity known as Gibbs free energy:
start text, G, i, b, b, s, space, f, r, e, e, space, e, n, e, r, g, y, end text, equals, start text, G, end text, equals, start text, H, end text, minus, start text, T, S, end text
where start text, H, end text is enthalpy, start text, T, end text is temperature (in kelvin, start text, K, end text), and start text, S, end text is the entropy. Gibbs free energy is represented using the symbol start text, G, end text and typically has units of start fraction, start text, k, J, end text, divided by, start text, m, o, l, negative, r, x, n, end text, end fraction.
When using Gibbs free energy to determine the spontaneity of a process, we are only concerned with changes in start text, G, end text, rather than its absolute value. The change in Gibbs free energy for a process is thus written as delta, start text, G, end text, which is the difference between start text, G, end text, start subscript, start text, f, i, n, a, l, end text, end subscript, the Gibbs free energy of the products, and start text, G, end text, start subscript, start text, i, n, i, t, i, a, l, end text, end subscript, the Gibbs free energy of the reactants.
delta, start text, G, end text, equals, start text, G, end text, start subscript, start text, f, i, n, a, l, end text, end subscript, minus, start text, G, end text, start subscript, start text, i, n, i, t, i, a, l, end text, end subscript
For a process at constant start text, T, end text and constant start text, P, end text, we can rewrite the equation for Gibbs free energy in terms of changes in the enthalpy (delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript) and entropy (delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript) for our system:
delta, start text, G, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, equals, delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, minus, start text, T, end text, delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript
You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for delta, start text, H, end text and delta, start text, S, end text are for the system of interest. This equation is exciting because it allows us to determine the change in Gibbs free energy using the enthalpy change, delta, start text, H, end text, and the entropy change , delta, start text, S, end text, of the system. We can use the sign of delta, start text, G, end text to figure out whether a reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium.
  • When delta, start text, G, end text, is less than, 0, the process is exergonic and will proceed spontaneously in the forward direction to form more products.
  • When delta, start text, G, end text, is greater than, 0, the process is endergonic and not spontaneous in the forward direction. Instead, it will proceed spontaneously in the reverse direction to make more starting materials.
  • When delta, start text, G, end text, equals, 0, the system is in equilibrium and the concentrations of the products and reactants will remain constant.

Calculating change in Gibbs free energy

Although delta, start text, G, end text is temperature dependent, it's generally okay to assume that the delta, start text, H, end text and delta, start text, S, end text values are independent of temperature as long as the reaction does not involve a phase change. That means that if we know delta, start text, H, end text and delta, start text, S, end text, we can use those values to calculate delta, start text, G, end text at any temperature. We won't be talking in detail about how to calculate delta, start text, H, end text and delta, start text, S, end text in this article, but there are many methods to calculate those values including:
When the process occurs under standard conditions (all gases at 1, start text, b, a, r, end text pressure, all concentrations are 1, start text, M, end text, and start text, T, end text, equals, 25, degrees, start text, C, end text), we can also calculate delta, start text, G, end text using the standard free energy of formation, delta, start subscript, f, end subscript, start text, G, end text, degrees.
Problem-solving tip: It is important to pay extra close attention to units when calculating delta, start text, G, end text from delta, start text, H, end text and delta, start text, S, end text! Although delta, start text, H, end text is usually given in start fraction, start text, k, J, end text, divided by, start text, m, o, l, negative, r, e, a, c, t, i, o, n, end text, end fraction, delta, start text, S, end text is most often reported in start fraction, start text, J, end text, divided by, start text, m, o, l, negative, r, e, a, c, t, i, o, n, end text, dot, start text, K, end text, end fraction. The difference is a factor of 1000!!

When is delta, start text, G, end text negative?

If we look at our equation in greater detail, we see that delta, start text, G, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript depends on 3 values:

delta, start text, G, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, equals, delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, minus, start text, T, end text, delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript
  • the change in enthalpy delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript
  • the temperature start text, T, end text
  • the change in entropy delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript
Temperature in this equation always positive (or zero) because it has units of start text, K, end text. Therefore, the second term in our equation, start text, T, end text, delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, will always have the same sign as delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript. We can make the following conclusions about when processes will have a negative delta, start text, G, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript:
  • When the process is exothermic (delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is less than, 0), and the entropy of the system increases (delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is greater than, 0), the sign of delta, start text, G, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript is negative at all temperatures. Thus, the process is always spontaneous.
  • When the process is endothermic, delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is greater than, 0, and the entropy of the system decreases, delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is less than, 0, the sign of delta, start text, G, end text is positive at all temperatures. Thus, the process is never spontaneous.
For other combinations of delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript and delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, the spontaneity of a process depends on the temperature.
  • Exothermic reactions (delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is less than, 0) that decrease the entropy of the system (delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is less than, 0) are spontaneous at low temperatures.
  • Endothermic reactions (delta, start text, H, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is greater than, 0) that increase the entropy of the system (delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, is greater than, 0) are spontaneous at high temperatures.
Can you think of any reactions in your day-to-day life that are spontaneous at certain temperatures but not at others?

Example 1: Calculating delta, start text, G, end text for melting ice

Three melting ice cubes in a puddle of water on a mirrored surface.
At what temperatures (if any) is the melting of ice a spontaneous process? Photo of ice cubes from flickr, CC BY 2.0.
Let's consider an example that looks at the effect of temperature on the spontaneity of a process. The enthalpy of fusion and entropy of fusion for water have the following values:
delta, start subscript, start text, f, u, s, end text, end subscript, start text, H, end text, equals, 6, point, 01, start fraction, start text, k, J, end text, divided by, start text, m, o, l, negative, r, x, n, end text, end fraction
delta, start subscript, start text, f, u, s, end text, end subscript, start text, S, end text, equals, 22, point, 0, start fraction, start text, J, end text, divided by, start text, m, o, l, negative, r, x, n, end text, dot, start text, K, end text, end fraction
What is delta, start text, G, end text for the melting of ice at 20, degrees, start text, C, end text?
The process we are considering is water changing phase from solid to liquid:
start text, H, end text, start subscript, 2, end subscript, start text, O, end text, left parenthesis, s, right parenthesis, right arrow, start text, H, end text, start subscript, 2, end subscript, start text, O, end text, left parenthesis, l, right parenthesis
For this problem, we can use the following equation to calculate delta, start text, G, end text, start subscript, start text, r, x, n, end text, end subscript:
delta, start text, G, end text, equals, delta, start text, H, end text, minus, start text, T, end text, delta, start text, S, end text
Luckily, we already know delta, start text, H, end text and delta, start text, S, end text for this process! We just need to check our units, which means making sure that entropy and enthalpy have the same energy units, and converting the temperature to Kelvin:
start text, T, end text, equals, 20, degrees, start text, C, end text, plus, 273, equals, 293, start text, K, end text
If we plug the values for delta, start text, H, end text, start text, T, end text, and delta, start text, S, end text into our equation, we get:
ΔG=ΔHTΔS=6.01kJmol-rxn(293K)(0.022kJmol-rxnK)=6.01kJmol-rxn6.45kJmol-rxn=0.44kJmol-rxn\begin{aligned} \Delta \text G &= \Delta \text H - \text{T}\Delta \text S \\ \\ &= 6.01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(293\,\cancel{\text K})(0.022\,\dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \\ \\ &= 6.01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-6.45\, \dfrac{\text{kJ}}{\text{mol-rxn}}\\ \\ &= -0.44 \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}
Since delta, start text, G, end text is negative, we would predict that ice spontaneously melts at 20, degrees, start text, C, end text. If you aren't convinced that result makes sense, you should go test it out!
Concept check: What is delta, start text, G, end text for the melting of ice at minus, 10, degrees, start text, C, end text?

Other applications for delta, start text, G, end text: A sneak preview

Being able to calculate delta, start text, G, end text can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction a reaction will proceed at a particular temperature, especially if we are trying to make a particular product. Chances are we would strongly prefer the reaction to proceed in a particular direction (the direction that makes our product!), but it's hard to argue with a positive delta, start text, G, end text!
Thermodynamics is also connected to concepts in other areas of chemistry. For example:

Summary

  • The second law of thermodynamics says that the entropy of the universe always increases for a spontaneous process: delta, start text, S, end text, start subscript, start text, u, n, i, v, e, r, s, e, end text, end subscript, equals, delta, start text, S, end text, start subscript, start text, s, y, s, t, e, m, end text, end subscript, plus, delta, start text, S, end text, start subscript, start text, s, u, r, r, o, u, n, d, i, n, g, s, end text, end subscript, is greater than, 0
  • At constant temperature and pressure, the change in Gibbs free energy is defined as delta, start text, G, end text, equals, delta, start text, H, end text, minus, start text, T, end text, delta, start text, S, end text.
  • When delta, start text, G, end text is negative, a process will proceed spontaneously and is referred to as exergonic.
  • Depending on the signs of delta, start text, H, end text and delta, start text, S, end text, the spontaneity of a process can change at different temperatures.

Try it!

For the following reaction, delta, start text, H, end text, start subscript, start text, r, x, n, end text, end subscript, equals, minus, 120, start fraction, start text, k, J, end text, divided by, start text, m, o, l, negative, r, x, n, end text, end fraction and delta, start text, S, end text, start subscript, start text, r, x, n, end text, end subscript, equals, minus, 150, start fraction, start text, J, end text, divided by, start text, m, o, l, negative, r, x, n, end text, dot, start text, K, end text, end fraction:
2, start text, N, O, end text, left parenthesis, g, right parenthesis, plus, start text, O, end text, start subscript, 2, end subscript, left parenthesis, g, right parenthesis, right arrow, 2, start text, N, O, end text, start subscript, 2, end subscript, left parenthesis, g, right parenthesis
At what temperatures will this reaction be spontaneous?
Note: Remember that we can assume that the delta, start text, H, end text and delta, start text, S, end text values are approximately independent of temperature.
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • starky ultimate style avatar for user natureforever.care
    Well I got what the formula for gibbs free energy is. but what's the nature of this energy and why is it called 'free'? It does free work is what textbooks say but didn't get the intuitive feel.
    (22 votes)
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    • piceratops seed style avatar for user RogerP
      The word "free" is not a very good one! In fact, IUPAC recommend calling it Gibbs energy or the Gibbs function, although most chemists still refer to it as Gibbs free energy.

      Gibbs originally called it available energy and that is a good term because it is the energy associated with a chemical reaction that is available (or you could say free) to do work, assuming constant T and P.
      (27 votes)
  • aqualine ultimate style avatar for user Ben Alford
    Is there a difference between the notation ΔG and the notation ΔG˚, and if so, what is it?
    (10 votes)
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    • blobby green style avatar for user dmelby
      STP is not standard conditions. Standard conditions are 1.0 M solutions and gases at 1.0 atm. Standard conditions does not actually specify a temperature but almost all thermodynamic data is given at 25C (298K) so many people assume this temperature.
      (4 votes)
  • blobby green style avatar for user ila.engl
    Hey I´m stuck: The ∆G in a reaction is negative but the ∆H was positive and it is assumed that a change temperature doesn´ t significantly affect entropy and entalpy. What does this do to 1) spontanity 2) spontanity at high temp 3) value or sign of ∆S

    i read it 3 times now but i´m still insecure - :(
    (3 votes)
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    • orange juice squid orange style avatar for user awemond
      This looks like a homework question, so I'll give you some hints to get you on the riht path rather than answering directly.

      3) We know that ∆G = ∆H - T∆S. Solving for ∆S, we have:

      ∆S=(∆H-∆G)/T.

      We know (from the question) that ∆G is negative and that ∆H is positive. Temperature is always positive (in Kelvin). From these values, we can know for certain whether ∆S is positive or negative (hint: remember that we are subtracting ∆G!).

      1) Knowing the sign of ∆G is enough to say whether the reaction is spontaneous or not under these conditions. If ∆G is negative (from the question), is the reaction spontaneous or non-spontaneous?

      2) Let's use ∆G = ∆H - T∆S again. Since ∆H and ∆S don't change significantly with temperature (given in the question), we can assume that they keep the same signs and values: i.e. ∆H is still positive and ∆S is still whatever sign you figured out above. As T increases, the T∆S component gets bigger. T is always positive, so if ∆S is positive then a bigger T∆S will make ∆G more negative (since we subtract T∆S). If ∆S is negative, then the negative signs (from the subtraction and the sign of ∆S) will cancel out, and so as T∆S gets bigger, ∆G will get more positive.
      (12 votes)
  • blobby green style avatar for user Phoebe Hall
    In the subject heading, 'When is ΔG is negative?', is it a typo that it says
    'When the process is endothermic, ΔHsystem > 0, and the entropy of the system decreases, ΔSsystem>0, the sign of ΔG is positive at all temperatures. Thus, the process is never spontaneous' shouldn't the entropy be < 0? if there is a decrease in entropy?
    (4 votes)
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  • leaf green style avatar for user Jasgeet Singh
    The Entropy change is given by Enthalpy change divided by the Temperature. Then how can the entropy change for a reaction be positive if the enthalpy change is negative?
    (4 votes)
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  • male robot donald style avatar for user Kaavinnan Brothers
    Hi all, Sal sir said we would prefer the reaction to proceed in a particular direction (the direction that makes our product!), but it's hard to argue with a positive ΔG! ( located before summary at other applications of del G) .can anybody please explain?
    (2 votes)
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  • blobby green style avatar for user izzahsyamimi042
    can an exothermic reaction be a not spontaneous reaction ?
    (1 vote)
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  • mr pants teal style avatar for user 1448169
    how do i see the sign of entropy when both reactant and product have the same phase
    (2 votes)
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    • hopper happy style avatar for user Stephen R. Collier
      We have to look up the ΔS for the whole reaction in a table (or test the reaction ourselves... I'd rather look it up!). The value will be either positive or negative. If the reaction can result in a phase change then we might be lucky enough to find a list that has the reaction with reactant and products in the phases we need.

      Otherwise we could calculate the change in energy and the use the specific heat equations to see if the phase would change. The example above with melting ice looks a little different because the reaction was a phase change (ice to water) instead of the usual combining or splitting of molecules.
      (1 vote)
  • duskpin seed style avatar for user estella.matveev
    Hi, could someone explain why exergonic reactions have a negative Gibbs energy value? I get it in terms of doing the calculations by looking at the graphs, but don't get it in terms of particles gaining or losing energy.
    (2 votes)
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    • leaf green style avatar for user Oliver McCann
      According to the laws of thermodynamics, ever spontaneous process will result in an increase in entropy and thus a loss in "usable" energy to do work. When an exergonic process occurs, some of the energy involved will no longer be usable to do work, indicated by the negative Gibbs energy.
      (1 vote)
  • starky ultimate style avatar for user William Shiuk
    Is there a differnce between "exothermic"-"exogoric" and "endothermic"-"endorgonic"?
    (1 vote)
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    • leaf red style avatar for user Richard
      Exothermic and endothermic refers to enthalpy; a negative or positive enthalpy respectively. Exergonic and endergonic refers to Gibbs free energy; a negative or positive value respectively.

      Hope that helps.
      (2 votes)