Final answer:
The radius of a hydrogen atom in the first excited state (n=2) according to Bohr's theory is calculated using the formula r_n = n² × aB, where the Bohr radius (aB) is approximately 0.529 × 10⁻¹° m, resulting in r_2 = 2.116 × 10⁻¹° m.
Step-by-step explanation:
Calculating the Radius of a Hydrogen Atom in the n=2 State
To calculate the radius of a hydrogen atom's electron orbit in the first excited state (n=2) using Bohr's theory, we must recall that the orbital radius is proportional to n². According to Bohr's model, the formula for the radius of an electron orbit in a hydrogen-like atom is given by:
r_n = n² × aB,
where aB is the Bohr radius and n is the principal quantum number. The Bohr radius, aB, has a value of approximately 0.529 × 10⁻¹° m. Therefore, for the first excited state where n=2, the radius would be:
r_2 = 2² × 0.529 × 10⁻¹° m,
r_2 = 4 × 0.529 × 10⁻¹° m,
r_2 = 2.116 × 10⁻¹° m.
This value represents the radius of an electron orbit of a hydrogen atom in the n=2 state according to Bohr's model.