The Bohr radius of the hydrogen atom is 0.0529 nm, or 0.529 angstrom. That’s the radius in the n=1 state. Approximately what is the radius of the hydrogen atom in the n=2 state? Pick the closest value since we have rounded off answers here.
0.05 nm
0.5 nm
0.2 nm
0.1 nm
The Correct Answer and Explanation is:
The correct answer is: 0.2 nm
Explanation
The Bohr model of the hydrogen atom describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits. One of the key results from the Bohr model is the expression for the radius of these orbits (also called “Bohr radii”) for different energy levels, given by the formula: rn=n2⋅a0r_n = n^2 \cdot a_0
Where:
- rnr_n is the radius of the orbit in the nn-th energy level,
- nn is the principal quantum number (n = 1, 2, 3, …),
- a0a_0 is the Bohr radius, approximately 0.0529 nm or 0.529 Å.
In the ground state (n=1n = 1), the radius of the hydrogen atom is r1=12⋅a0=0.0529r_1 = 1^2 \cdot a_0 = 0.0529 nm.
For the n=2n = 2 state: r2=22⋅a0=4⋅0.0529 nm=0.2116 nmr_2 = 2^2 \cdot a_0 = 4 \cdot 0.0529\, \text{nm} = 0.2116\, \text{nm}
This value rounds to 0.2 nm, which is the closest answer from the given options.
Why This Happens
In the Bohr model, as electrons jump to higher energy levels (higher nn), they occupy orbits that are farther from the nucleus. The radius increases with the square of nn, meaning the electron in the n=2n = 2 level is four times farther from the nucleus than in the ground state. This model helps explain spectral lines in hydrogen and was foundational in early quantum theory.
Although the Bohr model has been superseded by quantum mechanics, it still gives reasonably accurate estimates for hydrogen and hydrogen-like atoms, especially for explaining the relationship between energy levels and radii.
Hence, the radius of hydrogen in the n=2n = 2 state is approximately 0.2 nm.
