How the radius of hydrogen-like atoms grows with the principal quantum number n

Picture the electron in a hydrogen-like atom as a tiny traveler circling a bright nucleus. The distance it keeps—this orbit radius—depends on how high the energy level is. In the world of quantum ideas, that height is encoded in a number called the principal quantum number, n. And the math behind it has a simple, almost geometric sense: as n goes up, the orbit gets bigger in a predictable way.

A quick map of the idea

• What we mean by “hydrogen-like”: one electron orbiting a nucleus with charge +Ze. Think hydrogen (Z = 1) or ions that are basically hydrogen’s cousin, with a bigger nucleus.

• The Bohr radius, a0, is the natural length scale here. It’s about 0.529 angstroms ( Å), a hush-quiet, almost everyday scale in atoms.

• The radius r of the nth orbit in a hydrogen-like atom is r_n = a0 n^2 / Z. In words: it grows with the square of n, and it shrinks as the nuclear charge Z grows (so a bigger nucleus pulls the electron a bit closer).

So, for hydrogen itself (Z = 1), that tidy relationship becomes r_n = a0 n^2 = 0.529 Å × n^2. The key takeaway is simple: the radius grows with the square of n.

Let’s poke at the options you might see and why one is the right fit

• Option A looks like a Bohr-radius-like expression, but it’s got a different mass term and constants sprinkled in. It hints at the same spirit—r ∝ n^2—but it isn’t the clean, standard form we use in introductory crystallizations of the Bohr model.

• Option B, r = n² × 0.529 Å, is the clean hydrogen result: for hydrogen-like atoms with Z = 1, the radius scales as n² times the Bohr radius. This is the answer you’d give if you’re talking about hydrogen itself or using the simplest, commonly taught form for the radius.

• Option C, λ = h / √(2 e m V), looks like a wave- or momentum-related expression, but it’s not the radius story here. It mixes constants in a way that doesn’t describe how the electron’s orbit size depends on n.

• Option D, E = -13.6 × (Z² / n²) × (μ / m_e), is an energy formula. It tells you about the energy levels, not the orbit’s radius. So while it’s fundamental to understanding hydrogen-like systems, it answers a different question.

The physics in plain terms

Why does the radius depend on n in a quadratic way? In the Bohr picture, the electron’s angular momentum is quantized: L = nħ. The electron travels in a circular orbit, so its circumference 2πr must accommodate an integer number of de Broglie wavelengths. When you blend these ideas with the Coulomb attraction from the nucleus, you land on a neat result: r grows with n^2 for a fixed Z.

That “for a fixed Z” clause is important. If you crank up Z, the pull toward the nucleus strengthens. The same quantum conditions push the electron into a tighter orbit, so the radius actually gets smaller. In the general formula r_n = a0 n^2 / Z, you can see that trade-off clearly: bigger Z means smaller radius, all else equal.

A practical note on masses and what’s kept constant

In the real world, the mass that matters in the Bohr radius is the reduced mass μ of the electron-nucleus system. For hydrogen, μ is very close to the electron mass m_e, so the standard a0 ≈ 0.529 Å is a solid guide. For heavier nuclei, μ is a touch different, and that nudges the radius a bit. In classroom exercises, we often drop that small correction and keep μ ≈ m_e, which is why the simple r_n = a0 n^2 / Z formula feels so handy.

Why this is relevant in the NEET physics landscape

• It reinforces a core idea: quantum numbers aren’t just abstract labels. They map directly to measurable properties—the size of the electron’s orbit, the energy you’d assign to that level, and how tightly the electron hangs around the nucleus.

• It links two big concepts in one breath: the Bohr model’s elegance and the reality that hydrogen-like systems scale in a clean, predictable way. That clarity is exactly what makes these questions a good litmus test for understanding.

• It also offers a gentle bridge to more complex atoms. Once you’re comfortable with r_n ∝ n^2 / Z for a single-electron system, you can start spotting why multi-electron atoms behave differently, how shielding modifies the effective nuclear charge, and why orbital sizes still roughly track quantum numbers.

A little intuition you can carry around

Imagine climbing in a building where each floor corresponds to a quantum number n. As you go up, the distances from the center “nucleus” grow. It’s like the vibe changes from a snug studio apartment to a loft with a window that looks out over a bigger universe. It’s not that the electron becomes something else; it’s that the wave-like nature and the allowed quantum states let it reside farther out as you climb.

Tying it back to the bigger picture

• The Bohr radius isn’t just a number; it’s a scaling factor. It anchors the size of the world where the electron roams. When you multiply by n², you’re saying, “the size expands quadratically as you hop to higher energy levels.”

• The hydrogen-like statement r_n = a0 n^2 / Z shows the tug-of-war between the electron’s wish to be farther out and the nucleus’s pull. Larger Z tightens the orbit; higher n loosens it.

• If you ever see a question about the radius in a hydrogen-like system, expect the n² relation to appear somewhere in the answer, especially for the simplest case where Z = 1.

A few friendly reminders as you study

• Remember the language: “radius grows with the square of n” is the simple, memorable rule for hydrogen-like atoms when Z = 1.

• If Z isn’t 1, scale down by Z: r_n = a0 n^2 / Z. It’s a tiny tweak, but it changes the numbers a lot when you’re comparing hydrogen to its cousins.

• Keep the Bohr radius in your toolkit. It’s the bridge between the abstract quantum numbers and something you can actually picture or measure.

A closing thought

The radius-versus-n story is one of those crisp little chapters that makes quantum ideas feel approachable. It’s not just about memorizing a formula; it’s about recognizing a pattern: larger quantum numbers echo outward into bigger, more expansive orbits, and the nucleus keeps its grip with Z. When you see a problem ask for the radius of a hydrogen-like system, you now have a clean lens: r_n = a0 n^2 / Z, and for hydrogen itself, r_n = 0.529 Å × n^2.

If you’d like, I can walk through a quick example with a specific Z, say Z = 2 (a helium ion with one electron) to show how the radius shifts, or tease apart how the different formulas connect when you move from radius to energy. Either way, you’ll keep the thread: the n² growth is the heartbeat of the hydrogen-like radius story.