Understanding the Bohr radius for hydrogen-like atoms and how r scales with n and Z

Outline:

• Opening hook: picture the atom as a tiny solar system; radius is the size of the orbit.

• The star of the show: the radius formula for hydrogen-like atoms and what it tells us.

• Decoding the symbols: n, Z, h, ε0, m, e—and why each matters.

• Why this specific formula wins: its link to the Bohr radius and the n^2/Z scaling.

• What the other answer choices actually describe (and why they aren’t the radius formula here).

• A practical intuition: how radius changes with energy level and nuclear charge.

• Quick tips for solving radius-related questions in physics problems.

• Wrap-up: the big idea in one breath.

Hydrogen-like atoms and their "solar system" radius

If you’ve ever imagined electrons whizzing around a nucleus in neat, planet-like orbits, you’re not far off. In hydrogen-like atoms (that’s atoms with exactly one electron, like H, He+, Li2+, and so on), the distance from the nucleus to the electron in a given energy state is a well-defined number. And yes, there’s a clean formula that tells you that size.

The correct formula and what it means

The radius for an electron orbit in a hydrogen-like atom is

r = (n^2 h^2 ε0) / (π m Z e^2)

Here’s the scoop on each piece:

• n is the principal quantum number, the energy level the electron sits in. Higher n means the electron is farther out.

• h is Planck’s constant.

• ε0 is the permittivity of free space (a fundamental constant that tells you how strongly electric fields behave in empty space).

• m is the electron mass.

• Z is the atomic number—the number of protons in the nucleus, i.e., the charge pulling on the electron.

• e is the elementary charge (the charge of the electron).

A neat way to see what this means is to rewrite it in terms of a familiar landmark: the Bohr radius, a0. You can show that a0 = ε0 h^2 / (π m e^2). With that, the radius becomes:

r = n^2 a0 / Z

So, r grows with the square of the energy level (n^2) and shrinks as the nucleus pulls harder (1/Z). That means a He+ ion (Z = 2) has a smaller radius than a neutral hydrogen atom (Z = 1) in the same energy level, and if you excite the electron to a higher level, the radius balloons.

Why this formula matches the Bohr picture

The Bohr model isn’t the latest description of atomic structure in every fertilizer-for-thought sense, but it still gives a surprisingly accurate picture for single-electron systems. The Bohr radius a0 is the natural yardstick for the size of the electron’s orbit in the simplest cases. When you introduce Z, you’re basically saying, “more positive charge pulls harder, so the orbit shrinks.” That balance—n^2 up, Z up—captures the essence of how size scales in these systems.

What the other answer choices actually describe

Let’s peek at the tempting but not-quite-right options and see why they don’t give the radius in hydrogen-like atoms:

• B. R(1/n1^2 − 1/n2^2)

This looks like it could be tied to energy changes between levels, not a radius. It’s more aligned with energy spacing formulas where the difference in 1/n^2 terms shows up, rather than a direct spatial size.

• C. r = n^2 × 0.529 Å

This one nudges you toward the Bohr radius, but it’s missing the Z factor. It’s basically the hydrogen case for Z = 1, and it ignores the way the nucleus’ charge tightens the orbit. So it’s only correct in a very specific situation and would mislead you for ions with Z ≠ 1.

• D. KEmax = hν − φ

This is a classic from the photoelectric effect. It tells you about the maximum kinetic energy of emitted electrons when light shines on a surface, not about the radius of an electron’s orbit.

So the correct choice is indeed the r = (n^2 h^2 ε0) / (π m Z e^2). It’s the one that ties together the quantum level, the electron’s charge and mass, and the nuclear charge to give a size.

A little intuition you can carry to the next problem

Think of radius as a reflection of two opposing forces: the electron’s tendency to fly outward due to its energy, and the nucleus’s pull inward due to the positive charge. Increase n, and the energy climbs; the electron can sit farther away, so the orbit grows. Increase Z, and the nucleus pulls harder; even at the same energy level, the electron is squeezed closer in. That’s the elegant tug-of-war that the formula encodes.

A quick mental model for solving related questions

• If you’re asked to compare radii for the same n across ions, just compare Z. The bigger Z, the smaller the radius.

• If n changes but Z stays the same, radius grows as n^2. It’s a simple quadratic relationship—one more energy level makes a big difference.

• If you’re unsure whether a given problem is about a hydrogen-like system, check whether there’s only one electron and if the nucleus has a net charge Z > 0. If yes, the Bohr-type radius formula is your friend.

Tips to keep the math clear

• Remember the Bohr radius a0 is a composite constant: a0 = ε0 h^2 /(π m e^2). It’s handy to see how the r formula reduces to r = n^2 a0/Z.

• Watch the units. ε0 brings the right interplay of electric constants, while h, m, e carry the quantum and mass information. It’s easy to slip a unit on one piece and end up with a wrong radius.

• Don’t mix h with ħ by mistake.ħ = h/(2π); when you switch between expressions, the algebra has to carry that factor consistently.

Relating this idea to a broader physics thread

Radius isn’t just a number in an equation; it ties into spectroscopy, energy level spacings, and even how ions behave in plasmas or astrophysical environments. When you hear about spectral lines getting closer together for ions with higher Z, you’re really feeling the same physics from the other side—levels compressing as the nucleus pulls more strongly.

A touch of everyday metaphor to keep it memorable

Imagine you’re watching dancers on a stage. The principal steps (n) tell you how far from the spotlight the dancer is allowed to roam. The stage size (Z) makes the spotlight feel tighter or looser. The math is just those two ideas sitting together in a tidy formula, telling you exactly how big the “dance floor” is for the electron’s orbit.

Bringing it home

To recap: for hydrogen-like atoms, the radius of the electron’s orbit in a given energy level is

r = (n^2 h^2 ε0) / (π m Z e^2)

or, in the more intuitive Bohr-language, r = n^2 a0 / Z.

This compact relation tells you the size grows with the square of the energy level and shrinks with a stronger nuclear pull. The other choices either describe energy differences or are missing key pieces like Z, so they don’t capture the radius in these systems.

If you’re exploring these concepts further, you’ll find that the same thread—the way a system’s size responds to energy and charge—shows up in other quantum problems too. It’s one of those ideas that bridges simple models and the richer, more nuanced picture modern physics paints. And as you keep testing your understanding with slightly different questions, that bridge between intuition and formula becomes a lot more solid.