Ch.6 - Electronic Structure of Atoms

Problem 98

Chapter 6, Problem 98

Consider the discussion of radial probability functions in 'A Closer Look' in Section 6.6. (a) What is the difference between the probability density as a function of r and the radial probability function as a function of r ?

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The probability density function, often denoted as \( \psi^2 \), represents the probability per unit volume of finding an electron at a distance \( r \) from the nucleus.

The radial probability function, on the other hand, is the probability of finding an electron within a spherical shell at a distance \( r \) from the nucleus.

The radial probability function is obtained by multiplying the probability density by the surface area of the spherical shell, which is \( 4\pi r^2 \).

Thus, the radial probability function is given by \( 4\pi r^2 \psi^2 \), which accounts for the volume element in spherical coordinates.

In summary, while the probability density gives a local probability per unit volume, the radial probability function provides the total probability at a given radius, considering the entire spherical shell.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Density

Probability density refers to the likelihood of finding an electron in a specific region of space at a given distance from the nucleus. It is represented as a function of the radial distance 'r' and is calculated from the square of the wave function. This concept is crucial for understanding how electron distribution varies in an atom.

Radial Probability Function

The radial probability function describes the probability of finding an electron within a thin spherical shell at a distance 'r' from the nucleus. It is derived from the probability density and incorporates the volume of the spherical shell, which increases with 'r'. This function provides a more comprehensive view of electron distribution in three-dimensional space.

Spherical Coordinates

Spherical coordinates are a system of coordinates that define a point in three-dimensional space using three values: the radial distance from the origin, the polar angle, and the azimuthal angle. This system is particularly useful in quantum mechanics for describing the behavior of electrons in atoms, as it aligns with the symmetry of atomic orbitals.

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