To solve this problem, we will need to take a several steps.
Firstly, we need to understand that the location of an electron in an atom cannot be pinpointed exactly. Electrons exist in a cloud-like formation around the nucleus, where the cloud represents the probability of finding the electron at any given location. In quantum mechanics, this cloud is referred to as the electron cloud or orbital.
The electron's position in the orbital can be represented by a probability density function, ψ(r), and the probability density at any given point is proportional to the square of the wavefunction, |ψ(r)|².
For a hydrogen atom in the ground state, the radial probability density function is given by:
(1 / pi) * (1 / (bohr_radius)³) * exp(-2 * radius / bohr_radius)
where:
- The Bohr radius is defined as the most probable distance between the nucleus and the electron in a hydrogen atom which is approximately 0.529 Å.
- The radius of the sphere is half of 0.5 Å since the diameter is given which is 0.25 Å.
- The exp function is the mathematical constant e (approximately equal to 2.718) raised to the power of -2 * radius / bohr_radius.
Plugging in the values, we get a probability density of approximately 0.8356 for the ground state of a hydrogen atom.
However, the situation differs slightly when it comes to helium ion (He+). The helium ion has two protons in the nucleus, which effectively doubles the nuclear charge. This results in a stronger attraction between the nucleus and the electron, hence altering the way the electron behaves. This is accounted for by scaling the probability density by the square of the effective nuclear charge (Z²). For helium, Z = 2. Consequently, we multiply the probability density function of hydrogen by 4 (since Z² is 4 for helium). This gives a probability density for helium of about 3.3424.
The probability of finding the electron within a certain region is calculated by integrating the probability density over that region. Because we're looking for the probability within a sphere, we can multiply the probability density by the volume of the sphere.
The volume of a sphere is given by (4/3) * pi * (radius³), which in this case is approximately 0.0654 ų.
Therefore, by multiplying the helium probability density (3.3424) with the volume of the sphere (0.0654 ų), we obtain the desired probability, which is around 0.219.
Please note, this is a greatly simplified model which does not account for many effects such as electron correlation. For more accurate results, more sophisticated theoretical models or experimental measurements should be used.