Active mass of solids shouldn't necessarily be unity

Question

I've read that active mass of solids and pure liquids is considered unity. I'm absolutely sure there is some misunderstanding. I know that concentration of solids doesn't change during the course of reaction, but it shouldn't necessarily be unity either. Consider graphite for example. It's density is $2260\pu{g/L}$ $$\text{[C]}=\frac{n_C}{V_C}$$ where $n_C$ is the number of moles of the carbon sample and $V_C$ is the volume of that sample. $$\text{[C]}=\frac{w}{M\times V_C} $$ $$\text{[C]}=\frac{\rho_C}{M}=\frac{2260\pu{g/L}}{12\pu{g/mol}}\\=188.3\pu{mol/L}$$This concentration will be constant throughout a reaction if the phase doesn't change and temperature change is quite small. But this clearly is not unity. Then why is the concerntration of solids considered to be unity?

Answer 1

Rather, active mass of solids need not necessarily be unity, but arbitrarily is unity with advantage. You do not have to know the molar mass nor density.

The common thermodynamic expressions like $K = \frac{[C][D]}{[A][B]}$ or kinetic ones like $\frac{\mathrm{d}[C]}{\mathrm{d}t} = k \cdot [A] \cdot [B]$

should be more exactly written as $K = \frac{a_\mathrm{C}] \cdot a_\mathrm{D}}{a_\mathrm{A} \cdot a_\mathrm{B}}$ or $\frac{\mathrm{d}[C]}{\mathrm{d}t} = k \cdot a_\mathrm{A} \cdot \cdot a_\mathrm{B}$

where $a_\mathrm{X}$ is unitless quantity called activity, defined by the chemical potential of the substance $ \mu_\mathrm{A} = \left( \frac {\partial G}{\partial n_\mathrm{A}} \right)_{T,p}$ by formula:

$$\mu_\mathrm{A} = RT \ln {a_\mathrm{A, unitless}}$$

If we implement the standard chemical potential, we can define unitless activity as ratio of the activity with unit and the standard activity with the unit value.

$$\mu_\mathrm{A} = \mu_\mathrm{A}^{\circ} + RT \ln \left( \frac {a_\mathrm{A}}{a_\mathrm{A}^{\circ}} \right)$$

where $a_\mathrm{A}^{\circ}$ is the activity at the standard chemical potential $\mu_\mathrm{A}^{\circ}$. We can consider the activity $a_\mathrm{A} = x\ \text{"whatever unit"}$ and the standard activity $a_\mathrm{A}^{\circ} = 1 \ \text{"whatever unit"}$. If we arbitrarily choose $a^{\circ} = \pu{1 mol/L}$, then $a$ then more or less closely follow the respective concentration [A].

Chemical potential of the pure solid substance $\mu_\mathrm{A}^{\circ}$ does not depend on the amount of substance, so we arbitrarily choose $a^{\circ} = x_\mathrm{A}^{\circ}=1.0$ and $a$ then more or less follows the molar fraction of substance $x_\mathrm{A}$.

Answer 2

Let's start from the equilibrium constant, which may be defined $\ce{K=\frac{[C][D]}{[A][B]}}$. Of course, activities should be introduced here instead of concentration. But it is not the point here. The main use of this constant is to calculate how the composition of the system changes if at least one of the concentrations changes.

But if one of these concentrations does not change, whatever the perturbation of the system, it is no use maintaining it in the fraction defining $\ce{K}$. There would be two constants, one before and one after the sign "equal". It is much better to multiply or to divide the equilibrium constant by this concentration, so as to make it disappear from the fraction defining $\ce{K}$. For exemple if [B] does not change during a perturbation, it is much simpler to multiply $\ce{K}$ by $\ce{[B]}$ and find a new constant $\ce{K' = K·[B] = \frac{[C][D]}{[A]}}$, which is tabulated instead of $\ce{K}$ and can be used for describing any new change in the future composition of the system.

This is equivalent to saying that the concentration (or activity) [B| is chosen equal to $1$.