I am currently reading McQuarrie and Simon's Physical Chemistry and don't fully understand their discussion of standard states for activities. On page 990 they write (with a few minor adjustments to make it make sense):

If, on the other hand, one component is sparingly soluble in the other, then picking a standard state based upon Henry's Law instead of Raoult's law is more convenient. To see how we define the activity in this case, we start with $$\mu_j^\text{soln} = \mu_j^*+RT\ln\left(\frac{P_j}{P_j^*}\right)$$ (where the star refers to the quantities for the pure liquid, in this case chemical potential and vapor pressure). Because component $j$ is sparingly soluble, we use the fact that $P_j\to x_j k_{H,j}$ as $x_j\to 0$ where $k_{H,j}$ is the Henry's law constant of component $j$. If we substitute the limiting value $x_j k_{H,j}$ into the above equation for $P_j$, and obtain $$\mu_j^\text{soln} = \mu_j^*+RT\ln\left(\frac{x_j k_{H,j}}{P_j^*}\right)= \mu_j^*+RT\ln\left(\frac{k_{H,j}}{P_j^*}\right)+RT\ln{x_j}.$$ We define the activity of component $j$ by \begin{equation} \mu_j^\text{sln} = \mu_j^* + RT\ln\left(\frac{k_{H,j}}{P_j^*}\right)+RT\ln a_j \end{equation} so that $a_j\to x_j$ as $x_j\to 0$. The above expression becomes equivalent to $$\mu_j^\text{sln} = \mu_j^* + RT\ln{a_j}$$ if we define $a_j$ by $$a_j = \frac{P_j}{k_{H,j}}$$ and choose the standard state such that $$\mu_j^* = \mu_j^* + RT\ln\left(\frac{k_{H,j}}{P_j^*}\right)$$ or such that $k_{H,j} = {P_j^*}$. The standard state in this case requires that $k_{H,j} = P_j^*.$ This standard state may not exist in practice so it is called a hypothetical standard state. Nevertheless, the definition of activity involving Henry's law for dilute components is natural and useful.

The mathematical operations done here all make sense and I understand how they arrived at $$\mu_j^\text{sln} = \mu_j^* + RT\ln\left(\frac{k_{H,j}}{P_j^*}\right)+RT\ln a_j.$$ However, it is after this that I began to get confused. I don't understand what they mean by "and choose the standard state such that $$\mu_j^* = \mu_j^* + RT\ln\left(\frac{k_{H,j}}{P_j^*}\right)$$ or such that $k_{H,j} = {P_j^*}$." since $k_{H,j}$ and $P_j^*$ are properties of the substances involved and not necessarily as a state. Therefore, I don't see how fixing $k_{H,j} = {P_j^*}$ actually specifies a single standard state. Furthermore, it isn't clear why we can arbitrarily choose to have $k_{H,j} = {P_j^*}$ since the expression for chemical potential is already referenced to a standard state: namely the pure liquid $j$. Finally, it seems that the final expression for chemical potential $$\mu_j^\text{sln} = \mu_j^* + RT\ln{a_j}$$ is referenced to the pure liquid standard state chemical potential $\mu_j^*$ so I don't see where the "Henry's law standard state" actually factors into this equation. I apologize if this is a very basic question, but I would really appreciate if someone could clear up these confusions.



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