How do you 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.