In the first excited state of hydrogen atom, the energy of its electron is 10.2 eV. The radial distance of the electron from the hydrogen nucleus in this case is approximately: ____.

In the hydrogen atom the Bohr model gives the radius of the electron orbit for a principal quantum number $n$ as

$r\; =\; n²\; a₀$

where $a₀$ is the Bohr radius. Its measured value is

$a₀\; =\; 0.529 \AA \; =\; 5.29 \times 10⁻¹¹ m$

The first excited state corresponds to the second energy level, so $n\; =\; 2$. Substituting into the radius formula:

$r\; =\; (2)² a₀\; =\; 4 a₀$

$r\; =\; 4 \times 5.29 \times 10⁻¹¹ m\; =\; 2.116 \times 10⁻¹⁰ m$

Rounding to two significant figures gives

$r\; \approx \; 2.1 \times 10⁻¹⁰ m$

Therefore the radial distance of the electron from the nucleus in the first excited state is approximately 2.1 × 10⁻¹⁰ m.