# What's the difference between ΔG° and ΔG°'?

In biochemistry contexts, one often sees $\Delta G^{\circ\prime}$ ("delta G naught prime"), rather than the normal standard free energy change $\Delta G^{\circ}$ ("delta G naught").

What's the difference between the two quantities?
Is there a formal definition of the two terms?

• Can you give us a quote in context, perhaps from one of the books in which you've seen it? That would help in getting an answer, I think. Sep 13 '15 at 23:43
• I don't think it is a useful question for the simple reason similar notation nuances appears in all kinds of texts with wide variety of meaning.
– Greg
Sep 15 '15 at 12:42

The prime usually denotes a standard free energy that corresponds to an apparent equilibrium constant where the concentration (or activity) of one or more constituents is held constant.

For example, for $\ce{HA <=>A- + H+}$ the equilibrium constant is $K=\frac{[\ce{A-}][\ce{H+}]}{[\ce{HA}]}$ and the corresponding standard free energy change is $\Delta G^\circ=-RT \ln(K)$. If you know the value of $K$ and the start concentration of $\ce{HA}$ then you can compute the equilibrium concentrations of $\ce{HA}$, $\ce{A-}$, and $\ce{H+}$.

However, if the pH is held constant then $[\ce{H+}]$ is no longer a free variable and the apparent equilibrium constant is $K'=\frac{[\ce{A-}]}{[\ce{HA}]}$ and the corresponding standard free energy change is $\Delta G'^\circ=-RT \ln(K')$. So $\Delta G'^\circ=-RT \ln(K/[\ce{H+}])=\Delta G^\circ+RT\ln[\ce{H+}]$

In biochemistry there is often several important constituents in addition to $\ce{H+}$ that are held constant such as $\ce{Mg++}$, phosphate, etc.

• If you could list the constituents in biochemistry which are normally considered to have constant (unit) activity, that would improve the answer.
– R.M.
Sep 19 '15 at 17:01

Is there a formal definition of the two terms?

One form of the fundamental equation of thermodynamics is:

$$dU = TdS - P dV + \sum_{i}\mu_i dN_i$$

In this equation, the total internal energy has canonical variables $$V$$, $$S$$, and $$dN_i$$, where $$S$$ is the total entropy (in units of $$\frac{\mathrm J}{\mathrm K}$$), $$V$$ is the total volume, and $$N_i$$ is the number of moles of each molecular species present. $$T$$ is temperature; $$P$$ is pressure, and $$\mu_i$$ is the chemical potential of species $$i$$. The equation implies that if we were to know an equation that gave $$U$$ as a function of $$S$$, $$V$$, and all the $$N_i$$ we would know everything about the system. However, this is inconvenient for two reasons. First, $$S$$ and $$V$$ are extensive variables. Make the system bigger without changing its composition, and $$S$$ and $$V$$ increase. Second and more importantly, it is often difficult to hold $$S$$ constant when doing experiments. The same is true of $$V$$. (We live in a constant pressure atmosphere.)

Taking the Legendre transform of $$U$$ with respect to variables $$S$$ and $$V$$ gives a new fundamental equation:

$$dG = -S dT + V dP + \sum_{i}\mu_i dN_i$$

This equation means that if we knew a function that gave the Gibbs free energy as a function of $$T$$, $$P$$, and all the $$N_i$$, we could easily compute all thermodynamic properties of the system.

Say we're interested in the thermodynamics of ATP hydrolysis:

$$\ce{ATP + H2O <=> ADP + Pi}$$

This equation is really better written as

$$\ce{A-P3O10H3 + H2O -> A-P2O7H2 + H3PO4}$$

But of course in a buffer at pH 7, the conditions where many biochemical reactions occur, there really won't be $$\ce{H3PO4}$$ etc., there will be dissociation of protons $$\ce{H+}$$ and formation of anions like $$\ce{H2PO4-}$$ etc. So now all those reactions will have to be tracked too. The number of protons released by ATP is not the same as released by inorganic phosphate, and this is generally true. During a reaction, it is difficult to hold the number $$N_{\ce{H+}}$$ of protons constant, but through judicious choice of buffers etc. it is possible to hold the chemical potential of protons constant (i.e. do experiments at constant pH). Under such conditions, it makes sense to continue the Legendre transformations one step further:

$$dG^\prime = -S dT + V dP - N_{\ce{H+}} d\mu_{\ce{H+}} + \sum_{i \neq \ce{H+}}\mu_i dN_i$$

$$\Delta G^{\circ \prime}$$ is a Legendre transform of $$\Delta G^{\circ}$$ with respect to the number of protons in the system.

Robert Alberty's paper from 1994 is a good place for further reading.

• If I could accept two answers, I'd also accept yours for the Alberty reference, in particular as he apparently was the head of the IUPAC-IUBMB (International Union of Pure and Applied Chemistry-International Union of Biochemistry and Molecular Biology) panel on Biochemical Thermodynamics nomenclature (official panel recommendations), which also has a discussion of the issue.
– R.M.
Sep 19 '15 at 17:17
• @Curt F. I really like your answer since it takes into account of the Legendre transformation. Could you please elaborate on the intuition or motivation behind using such a transformation? Thanks for your answer. Jan 2 '21 at 0:54

The prime has nothing to do with whether a concentration is held constant. The prime is used to indicate that some species, typically protons or ions such as Mg, are being set to a value other than the standard 1M concentration for use in a reference free energy. Although the prime has been adopted by some in the biochemical community and was endorsed in 1994 by IUPAC, the prime is not used in other communities for good reason - it is much better to explicitly state what concentrations are being used for the reference free energy. When using a reference free energy other than the standard free energy, concentrations can vary or be held fixed. In most biochemical situations, it is useful to keep the pH fixed since buffers are used. But this is a separate issue.

The biochemistry convention does not assume all solutions are 1 M. If you did this then you would have [H+] = 1 M. This simply never happens in biochemical reactions. Instead, we assume pH = 7. We also assume that water has an activity of 1 even though its concentration is 55 M.

• The use of $\Delta G^\circ$ does not assume that all solutions are 1 M. It assumes that the standard state is 1 M ideal solution. Also, activity is always defined as $\gamma [X]/[X]^\circ$ which is always 1 for pure substances. Sep 15 '15 at 10:40

Delta G naught means that the reaction is under standard conditions (25 celsius, 1 M concentraion of all reactants, and 1 atm pressure). Delta G naught prime means that the pH is 7 (physiologic conditions) everything else is the same. The concentration of [H+] now isn't 1 molar because 1 molar concentration would be an extremely low pH (0). Delta G naught prime is just like Delta G naught but for biology.