What does chemical potential really mean?

Question

I do not completely understand the concept of chemical potential $\mu$:

It is usually defined by

$$\delta Q = T dS = dU + pdV + \mu dn$$

By adding, lets say 1 mole of mass to a gas of 10 moles ($\delta n = 1 mole$), I will have 11 moles afterwards. But what does it mean in that context to "add" some amount of a component? Finally, it must make a clear difference if this added 1 mole is "hot" or "cold" and so $\delta Q$ must somehow depend on a $\mu$ that also depends on T: $\mu = \mu(T)$. But what T? T is the temperature of my system not of what I try to add.

Moreover, to fully describe a process of "adding something of given temperature" one does not really need a new quantity $\mu$; the final state after mixing can be described uniquely by gas laws, etc... What is the benefit of defining a new quantity $\mu$ and what does it mean in view of my question?

Answer 1

The quote below is from this excellent tutorial Baierlein: The elusive chemical potential

Any response to the question, ‘‘What is the meaning of the chemical potential?,’’ is necessarily subjective. What satisfies one person may be wholly inadequate for another. Here I offer three characterizations of the chemical potential (denoted by $\mu$) that capture diverse aspects of its manifold nature.

(1) Tendency to diffuse. As a function of position, the chemical potential measures the tendency of particles to diffuse.

(2) Rate of change. When a reaction may occur, an extremum of some thermodynamic function determines equilibrium. The chemical potential measures the contribution (per particle and for an individual species) to the function’s rate of change.

(3) Characteristic energy. The chemical potential provides a characteristic energy: $(\partial E/\partial N)_{S,V}$ , that is, the change in energy when one particle is added to the system at constant entropy (and constant volume).

These three assertions need to be qualified by contextual conditions, as follows.

(a) Statement (1) captures an essence, especially when the temperature $T$ is uniform. When the temperature varies spatially, diffusion is somewhat more complex and is discussed briefly under the rubric ‘‘Further comments’’ in Sec. IV.

(b) Statement (2) is valid if the temperature is uniform and fixed. If, instead, the total energy is fixed and the temperature may vary from place to place, then $\mu/T$ measures the contribution. When one looks for conditions that describe chemical equilibrium, one may focus on each locality separately, and then the division by temperature is inconsequential.

(c) The system’s ‘‘external parameters’’ are the macroscopic environmental parameters (such as external magnetic field or container volume) that appear in the energy operator or the energy eigenvalues. All external parameters are to be held constant when the derivative in statement (3) is formed. The subscript $V$ for volume illustrates merely the most common situation. Note that pressure does not appear in the eigenvalues, and so—in the present usage—pressure is not an external parameter.

Answer 2

The chemical potential is not usually defined by the equation you gave. The chemical potential is defined as the partial molar Gibbs free energy of a chemical species in solution (or a pure species).