Final answer:
The radius of the first orbit of a helium ion is smaller than that of a hydrogen atom due to the stronger nuclear charge in helium, which pulls the electron closer. Both radii can be calculated using the Bohr model and the Bohr radius.
Step-by-step explanation:
To calculate the radius of the first orbit for a hydrogen atom and a helium ion, we use the Bohr model of the atom. The Bohr model gives us a formula for the radius of the n-th orbit of a hydrogen-like atom (an atom with only one electron, like He+), which is rn = n2a0/Z, where n is the principal quantum number, a0 is the Bohr radius (5.292 x 10-11 m), and Z is the atomic number of the element.
For a hydrogen atom (Z = 1) in its first orbit (n = 1), the radius is simply the Bohr radius: r1 = a0. For helium ion (He+, Z = 2) in its first orbit, the radius would be half the Bohr radius: r1 = a0/2. Therefore, the radius of the first orbit of helium ion is smaller than that of hydrogen.
This can be explained by considering that the nucleus of helium has a stronger positive charge than hydrogen, which pulls the single electron of the helium ion closer to the nucleus, resulting in a smaller radius for its orbit.