Question

The radius r of a hydrogen atom (which consists of a proton and a comparatively lightweight electron) can only depend on the (equal) magnitudes e of the proton's and the electron's charge, the Coulomb constant 1/4 πϵ₀ that characterizes the strength of the electrostatic attraction between the proton and electron, the mass m of the orbiting electron, and (because quantum mechanics is likely involved) Planck's constant in the form h/2 π = h. (Since the proton remains essentially at rest, its mass turns out to be irrelevant.)

a. Assuming that r depends only on the stated quantities, use dimensional analysis and the information given inside the front cover of the text to find a plausible formula for the radius of a hydrogen atom.
b. Calculate a numerical estimate of the radius.
c. Explain why this is only an estimate.

Answer 1

Dimensional analysis for the hydrogen atom's radius relies on a formula involving Planck's constant, the electron's charge and mass, and the Coulomb constant. For the ground state of the atom, the radius equals the Bohr radius, which is a theoretical value derived from Bohr's model of the atom. This value is an estimate because Bohr's model is an approximation and real atomic structure is better described by quantum mechanics.

Step-by-step explanation:

a) Dimensional Analysis to Find Formula for Hydrogen Atom's Radius:

According to dimensional analysis, the radius r of a hydrogen atom can only depend on the magnitude of charge e, the Coulomb constant k, the mass m of the electron, and Planck's constant h. These quantities can be combined in such a way to ensure that their dimensions form a length (L), since r has the dimension of length. The plausible formula for the radius based on these quantities should be proportional to some power of these quantities that results in the dimension L. Based on the classical derivation of Bohr's model, the radius r is given by the Bohr radius:

r = n² × h² / (4π² × m × e² × k)

For the ground state (n=1),

r = h² / (4π² × m × e² × k)

b) Calculation of Hydrogen Atom Radius Using Given Values

To provide a numerical estimate of the radius (Bohr radius), we can substitute the Planck constant (h), electron mass (m), elementary charge (e), and Coulomb's constant (k) values into the above formula. However, the question seems to have a confusion about the value of k. The given value k = 6.67 x 10⁻¹¹N·m²/C² is actually the gravitational constant, not the Coulomb constant. Assuming the force between the electron and proton remains constant, the radius wouldn't change.

c) Why This is Only an Estimate

This derived formula is based on Bohr's model, which makes assumptions that are only applicable under specific circumstances and the model is superseded by quantum mechanics. The actual distribution of the electron in the atom is not a simple orbit, as suggested by Bohr's model, but rather a probabilistic cloud described by quantum mechanics. The most probable radius derived from quantum mechanics corresponds closely to the Bohr radius but may differ slightly due to quantum mechanical corrections.