Question

Explain the significance of the fact that the peak in the radial probability distribution plot for the \(n=1\) level of an \(\mathrm{H}\) atom is at 0.529 A. Is the probability of finding an electron at 0.529 A from the nucleus greater for the \(1 s\) or the \(2 s\) orbital?

Short answer

The probability of finding an electron at 0.529 A is greater for the \(1s\) orbital.

Step-by-step solution

- Understand the given data

The problem states that the peak in the radial probability distribution plot for the \( n=1 \) level of a hydrogen atom is at 0.529 angstroms (A). This value is important to understand the likelihood of finding an electron at this distance from the nucleus.

- Recognize the significance

The value 0.529 A is known as the Bohr radius, which is the most probable distance for finding the electron in the ground state (\(1s\) orbital) of a hydrogen atom. This indicates that the highest probability of finding the electron occurs at this distance.

- Compare radial probability of \(1s\) and \(2s\) orbitals

For the \(1s\) orbital (\(n=1\)), the radial probability distribution has its maximum at the Bohr radius (0.529 A). For the \(2s\) orbital (\(n=2\)), while there might be a probability of finding an electron at 0.529 A, the radial probability distribution is different. It has additional nodes and its peak occurs at a different value, not at 0.529 A.

- Conclude the comparison

Given that the peak of the radial probability distribution for the \(2s\) orbital is not at 0.529 A, the probability of finding an electron at 0.529 A is highest for the \(1s\) orbital compared to the \(2s\) orbital.

Key concepts

Radial Probability Distribution

To understand the behavior of electrons in an atom, we often look at the radial probability distribution. This describes the likelihood of finding an electron at a certain distance from the nucleus. For each orbital, this probability varies and is shown as a plot. The peak of the radial probability distribution is particularly significant because it represents the most probable distance where an electron can be found.

In the case of a hydrogen atom, the peak in the radial probability distribution plot for the ground state (=1ot found at 0.529 A, which is also called the Bohr radius. This value is crucial when discussing the probability distribution around the nucleus.

Hydrogen Atom Orbitals

The concept of orbitals is essential in understanding electron behavior in hydrogen atoms. Each orbital is defined by a quantum number denoted by 'n'. These orbitals have different shapes and sizes, influenced by the quantum number:

• =1 is called the 1s orbital (Ground State)
• =2 is called the 2s orbital

For the 1s orbital in a hydrogen atom, also known as the ground state, the Bohr radius (0.529 A) represents the most probable distance from the nucleus where the electron can be found.

In contrast, the 2s orbital has a more complex radial probability distribution with additional nodes where the probability drops to zero. This highlights why electrons in higher orbitals have different probability distributions compared to those in the ground state.

Ground State Electron Probability

The ground state of a hydrogen atom is its simplest and most stable energy state. In this state, the electron is in the 1s orbital, and the probability of finding it is highest at the Bohr radius of 0.529 A. This radius is a fundamental characteristic of the hydrogen atom and gives a clear insight into its electronic structure.

Mathematically, we can express the radial probability distribution in the ground state as:

P(r) = |R_{1s}(r)|^2 4πr^2 ,

where P(r) is the probability of finding the electron at distance r , R_{1s}(r) is the radial wave function, and 4πr^2 is a volume element in spherical coordinates. Comparing the ground state and excited states like the 2s orbital reveals why the ground state offers a higher probability of finding the electron at the Bohr radius.