Animated Solution for Physics - Atoms and Nuclei: Imagine that the electron in a hydrogen atom is replaced by a muon ($\mu$). The mass of muon particle is $207$ times that of an electron and charge is equal to the charge of an electron. The ionisation potential of this hydrogen atom will be

Muonic Hydrogen and Ionization Potential

Imagine a standard hydrogen atom. We have a tiny, energetic electron orbiting a central proton. It is a delicate dance governed by electrostatic forces and quantum mechanics. Now, let's perform a fascinating thought experiment: what if we replace that electron with a muon?

A muon (${\mu }^{-}$) is essentially a heavier cousin of the electron. It carries the exact same negative charge, but it is a staggering $207$ times more massive. This single change in mass completely alters the energy landscape of the atom.

Mass

To understand how the ionization potential changes, we need to look at the master equation derived from Bohr's model. The ionization potential ($IP$) is the energy required to completely remove the orbiting particle from the ground state ($n=1$). The formula is given by:

Notice the variables in the numerator. The ionization potential is directly proportional to the mass ($m$) of the orbiting particle and the fourth power of its charge ($q^4$).

Since the muon has the exact same charge as an electron ($q_{\mu }=q_e$), the charge term remains constant. Therefore, the ionization potential becomes solely dependent on the mass:

To find the new ionization potential, we can set up a simple ratio comparing the muonic hydrogen atom to the standard hydrogen atom:

We know that the ionization potential of a standard hydrogen atom ($I{P}_e$) is a familiar $13.6\text{ eV}$. We are also given that the mass of the muon is $207$ times the mass of the electron ($m_{\mu }=207m_e$).

Substituting these known values into our ratio, we get:

The electron mass ($m_e$) cancels out beautifully, leaving us with a straightforward multiplication:

Because the muon is so much heavier, it is bound much more tightly to the nucleus. It requires a massive $2815.2\text{ eV}$ of energy to tear it away, compared to just $13.6\text{ eV}$ for an electron.

Radius and Velocity

This thought experiment opens the door to other interesting questions. How does the heavier mass affect the size of the atom?

According to Bohr's model, the radius of the orbit is inversely proportional to the mass ($r\propto \frac{1}{m}$). This means the muon orbits exactly $207$ times closer to the proton than an electron would! The muonic hydrogen atom is incredibly compact.

What about its speed? The velocity of the orbiting particle in Bohr's model is independent of its mass ($v\propto m^0$). Therefore, despite being $207$ times heavier and orbiting $207$ times closer, the muon travels at the exact same speed as an electron in a standard hydrogen atom. These proportionalities are powerful tools for solving complex atomic physics problems.