Final answer:
The expression for the radial distribution function for a 3p electron is complex and includes Laguerre polynomials, but the most probable radius is found by maximizing this function. For a hydrogen-like atom, this radius can be estimated to be 9 times the Bohr radius for a 3p electron, which would be approximately 476.1 picometers.
Step-by-step explanation:
The radial distribution function for a 3p electron in a hydrogenic atom with atomic number Z follows similar principles as the ground state of hydrogen but takes into account the higher energy level (n=3) and azimuthal quantum number (l=1). While the exact expression is complex, involving associated Laguerre polynomials, the main idea is to find the radius at which the electron is most likely to be found by maximizing the radial distribution function. This involves taking the derivative of the function with respect to r, setting it to zero, and solving for r. For the 3p orbital, there are two radial nodes, and the most probable radius will be greater than in the 1s orbital. The function's expression will depend on the Bohr radius and the principle quantum number n.
To find the most probable radius for a 3p electron, we apply the general formula that tells us the most probable radius in a hydrogen-like atom is proportional to n². Since we are dealing with a 3p electron (n=3), the most probable radius will be larger than that of the ground state of hydrogen. Using the proportionality to the square of the orbit's principal quantum number, we would expect the most probable radius for a 3p electron to be 9 times that of the 1s orbital, as n=3 for the 3p electron, as opposed to n=1 for the 1s electron. Assuming that Z is 1, as in a normal hydrogen atom, and the Bohr radius (a0) is approximately 52.9 picometers, the most probable radius of a 3p electron can be estimated as 52.9 pm * 9 or approximately 476.1 picometers.