In contexts, one regularly sees $\\Delta G^\\circ\\prime$ (\"delta G nothing prime\"), fairly than the normal standard free energy readjust $\\Delta G^\\circ$ (\"delta G naught\").

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What\"s the difference between the 2 quantities?Is over there a formal definition of the 2 terms?



The prime typically denotes a standard free energy that corresponds to an noticeable equilibrium constant where the concentration (or activity) that one or much more constituents is hosted constant.

For example, because that $\\ceHA A- + H+$ the equilibrium constant is $K=\\frac<\\ceA-><\\ceH+><\\ceHA>$ and also the equivalent standard totally free energy change is $\\Delta G^\\circ=-RT \\ln(K)$. If you know the value of $K$ and also the start concentration that $\\ceHA$ climate you deserve to compute the equilibrium concentration of $\\ceHA$, $\\ceA-$, and $\\ceH+$.

However, if the pH is held constant then $<\\ceH+>$ is no longer a totally free variable and the evident equilibrium consistent is $K\"=\\frac<\\ceA-><\\ceHA>$ and the matching standard complimentary energy adjust is $\\Delta G\"^\\circ=-RT \\ln(K\")$. So $\\Delta G\"^\\circ=-RT \\ln(K/<\\ceH+>)=\\Delta G^\\circ+RT\\ln<\\ceH+>$

In over there is often several essential constituents in addition to $\\ceH+$ that room held consistent such together $\\ceMg++$, phosphate, etc.

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answer Sep 15 \"15 at 10:33

january JensenJan Jensen
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Is there a formal an interpretation of the 2 terms?

One form of the basic equation of thermodynamics is:

$$dU = TdS - ns dV + \\sum_i\\mu_i dN_i$$

In this equation, the complete internal energy has canonical variables $V$, $S$, and $dN_i$, wherein $S$ is the full entropy (in devices of $\\frac\\mathrm J\\mathrm K$), $V$ is the full volume, and $N_i$ is the number of moles of every molecular types present. $T$ is temperature; $P$ is pressure, and also $\\mu_i$ is the chemical potential of species $i$. The equation suggests that if us were to recognize an equation that gave $U$ as a role of $S$, $V$, and also all the $N_i$ we would recognize everything around the system. However, this is inconvenient for two reasons. First, $S$ and $V$ are extensive variables. Make the mechanism bigger without changing its composition, and $S$ and also $V$ increase. 2nd and an ext importantly, the is often difficult to hold $S$ continuous when act experiments. The very same is true of $V$. (We live in a continuous pressure atmosphere.)

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

$$dG = -S dT + V dP + \\sum_i\\mu_i dN_i$$

This equation way that if us knew a function that gave the Gibbs totally free energy as a duty of $T$, $P$, and also all the $N_i$, we could easily compute every thermodynamic nature of the system.

Say we\"re interested in the thermodynamics the ATP hydrolysis:

$\\ceATP + H2O ADP + Pi$

This equation is really better written as

$\\ceA-P3O10H3 + H2O -> A-P2O7H2 + H3PO4$

But of food in a buffer in ~ pH 7, the problems where numerous biochemical reaction occur, there yes, really won\"t it is in $\\ceH3PO4$ etc., there will certainly be dissociation of proton $\\ceH+$ and also formation that anions favor $\\ceH2PO4-$ etc. So now all those reactions will have to be tracked too. The number of protons exit by ATP is not the same as released by not natural phosphate, and also this is normally true. During a reaction, that is daunting to host the number $N_\\ceH+$ of protons constant, yet through judicious choice of buffers etc. That is feasible to organize the chemical potential that protons continuous (i.e. Do experiments at consistent pH). Under such conditions, it makes sense to proceed the Legendre transformations one step further:

$$dG^\\prime = -S dT + V dP - N_\\ceH+ d\\mu_\\ceH+ + \\sum_i \\neq \\ceH+\\mu_i dN_i$$

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