It is clear how to measure thermodynamics quantities such as temperature, pressure, energy, particle number and volume. But I have no idea how to measure chemical potential.
Could someone please provide some examples of how one could measure the chemical potential?
We can't measure chemical potentials but that's OK because the actual value of the potential is not important, what matters is its difference from some other state. So, chemical potentials are compared. For example, in vapor-liquid equilibrium the chemical potential of the liquid component is equal to the chemical potential of the vapor component. If the vapor phase can be treated as an ideal gas we can calculate (not measure) its chemical potential. If it is not an ideal gas we need some other equation of state along with suitable assumptions about the interaction of components in order to do the calculation. It is also possible to calculate chemical potentials by computer simulation.
The general procedure to measure difference in the chemical potential is to compare it to some standard reference. Two standard references are in common use: ideal-gas state and ideal solution. In both cases the chemical potential of the reference state is $$ \mu_i = \mu_i^0 + RT \ln x_i $$ where $x_i$ is the mol fraction of the component and $\mu_i^0$ is the chemical potential of the pure component at the same temperature and pressure. One then calculates the departure of the chemical potential from the reference state using auxiliary properties such as activity coefficients, fugacity coefficients, etc.
You can measure it indirectly by using other extensive quantities and applying thermodynamic relations (see https://en.wikipedia.org/wiki/Table_of_thermodynamic_equations). For instance, you could use $$\mu = (\frac{\partial G}{\partial N})_{p,T}$$
As for measuring it directly, it is not possible to measure it directly.
You can check the answer in Is there a tool to measure the chemical potential of a system? for a reason about the last point.
It's possible to measure chemical potential directly. Chemical potential is exactly analogous to pressure, via the fundamental relation S(U, V, N).
While P is what is equalized under a moving wall, $\mu$ is what is equalized under a permeable wall.
While V is what changes to equalize P, N is what changes to equalize $\mu$.
We can measure P by placing it next to a reference system via a moveable wall and see if V increases, decreases, or stays the same. Then we say our system has smaller, greater, or equal pressure than the reference.
We can measure $\mu$ by placing it next to a reference system via a permeable wall and see if N increases, decreases, or stays the same. Then we say our system has smaller, greater, or equal chemical potential than the reference.
We will see the pressure of an ideal gas decreases linearly with N. Since the minimum N is zero, the corresponding minimal pressure of an ideal gas is that with zero N. So we then simply define zero pressure as the pressure of an empty container.
The $\mu$ of an ideal gas increases with ln(P), so it is undefined at zero P (it approaches negative infinity). So there's no clear system we should take as having zero $\mu$.
Nevertheless, we can measure the (relative) chemical potential of any system by this method. It is important that the other thermodynamic variables remain constant as we measure.
