reference states for activities

Question

I wanted to ask clarifications about a passage in Atkin's physical chemistry book, chapter 9, in the paragraph 9.2 description of equilibrium.

For studying equilibrium of a reaction, where $\nu_i$ are the stechiometric coefficients it imposes $\sum_i \nu_i \mu_i=0$ with $\mu$ the chemical coefficients. At this point it writes the chemical coefficients as a function of the activities $\mu_i=\mu_i^\circ+RT\ln(a_i)$ and identifies the term $\sum_i \nu_i \mu_i^\circ$ with the standard reaction Gibbs energy. I do not understand this identification. I thought that the standard state for the definition of activities (even dependent on wether the substance is a solute or a solvent) was not the same reference state for computing the reaction Gibbs energy (and does not depend on the role played by the substance in the reaction). I'm surely getting something wrong...

Answer 1

$\mu_i=\mu_i^o+RT ln(a_i)$ is the definition of activity ($a_i$).

Therefore, the standard state for standard chemical potential ($\mu_i^o$) is necessarily part of the definition of activity.

For a pure liquid or solid, the standard state is the pure substance at 1 bar.

For a liquid solvent, the standard state is also the pure liquid at 1 bar.

For a gas, the standard state is the fictitious state of an ideal gas at 1 bar.

For solutes, there are multiple conventions, for example the fictitious state of the solute having a molality of 1 mole/kilogram and exhibiting the behavior it does at infinite dilution.

$\Delta G^o = \sum_i \nu_i \mu_i^o$ by definition.

So there is no one standard state for $\Delta G^o$, because the various substances may have different standard states.

Answer 2

So is the point of contention here "how does the standard state for the activities relate to the standard state for the Gibbs free energy change"?

To see how choice of standard state for an activity relates to the standard state chosen for the chemical potential, write the equation as

$$\mu_i = \mu_i^\circ + R T \ln a_i = \mu_i^\circ + R T \ln \frac{\gamma_i c_i}{c_i^\circ}$$

where the standard states are consistent.The standard state is being defined so that when $\gamma_i c_i = c_i^\circ$, $\mu_i = \mu_i^\circ$. If we're using molarities, we can choose $c_i^\circ = 1\ M$.

Now suppose you want to define a standard state that's different from 1 in the units you're using for concentration; e. g, suppose we have a reaction that includes $\rm H^+$ and we want to define the biological standard state ($^\oplus$) as $c_{\rm H^+}^\oplus = 10^{-7}\ M = 10^{-7} c_{\rm H^+}^\circ$, while using molarities as our concentration unit.

We can compute a new chemical potential:

$$\begin{array}{rcl}\mu_{\rm H^+} &=& \mu_{\rm H^+}^\circ + R T \ln \dfrac{\gamma_i c_{\rm H^+}}{c_{\rm H^+}^\oplus}\\ &=& \mu_{\rm H^+}^\circ + R T \ln \dfrac{\gamma_i c_{\rm H^+}}{10^{-7} c_{\rm H^+}^\circ}\\ &=& (\mu_{\rm H^+}^\circ + 7 R T\ln 10) + R T \ln \dfrac{\gamma_i c_{\rm H^+}}{c_{\rm H^+}^\circ}\\ &=& \mu_{\rm H^+}^\oplus + R T \ln \dfrac{\gamma_i c_{\rm H^+}}{c_{\rm H^+}^\circ}\\ &=& \mu_{\rm H^+}^\oplus + R T \ln a_{\rm H^+} \end{array}$$

Now at unit activity $\gamma_i c_i = c_i^\circ$ and $\mu_i = \mu_i^\oplus$. Your chemical potential (and your Gibbs free energy changes) are now defined in terms of this new standard state. And note that the chemical potential at this new standard state ($\mu_{\rm H^+}^\oplus$) is indeed temperature dependent. To convert between the two standard state Gibbs free energies in this particular case,

$$\Delta G^\oplus = \Delta G^\circ + 7 \nu R T \ln 10$$

where $\nu$ is the stoichiometric coefficient of $\rm H^+$ in the reaction.

TL;DR: The standard state you choose for the concentration at unit activity becomes the standard state for the standard state Gibbs free energy change.