Question

Determine the radii of the first and second quantum orbits in the hydrogen atom.

Short answer

The radii of the first and second quantum orbits in the hydrogen atom are approximately equal to \(5.29 \times 10^{-11} m\) and \(2.12 \times 10^{-10} m\), respectively.

Step-by-step solution

Determine the radius of the first quantum orbit

For the first quantum orbit, the principal quantum number is n=1. Plugging this value into the formula, we get: \(r_1 = a_0 \times 1^2\)

Calculate the radius of the first quantum orbit

Now, substitute the value of the Bohr radius (\(a_0 = 5.29 \times 10^{-11} m\)): \(r_1 = (5.29 \times 10^{-11} m) \times 1^2\) \(r_1 = 5.29 \times 10^{-11} m\) So, the radius of the first quantum orbit is approximately equal to \(5.29 \times 10^{-11} m\).

Determine the radius of the second quantum orbit

For the second quantum orbit, the principal quantum number is n=2. Plugging this value into the formula, we get: \(r_2 = a_0 \times 2^2\)

Calculate the radius of the second quantum orbit

Now, substitute the value of the Bohr radius (\(a_0 = 5.29 \times 10^{-11} m\)): \(r_2 = (5.29 \times 10^{-11} m) \times 2^2\) \(r_2 = 5.29 \times 10^{-11} m \times 4\) \(r_2 = 2.12 \times 10^{-10} m\) So, the radius of the second quantum orbit is approximately equal to \(2.12 \times 10^{-10} m\). In conclusion, the radii of the first and second quantum orbits in the hydrogen atom are approximately equal to \(5.29 \times 10^{-11} m\) and \(2.12 \times 10^{-10} m\), respectively.

Key concepts

Quantum Orbits

The concept of quantum orbits is central to Bohr's Model of the atom. According to this model, electrons travel in fixed paths or orbits around the nucleus, much like planets around the sun. These paths are called quantum orbits.

Bohr introduced the idea that electrons could only occupy certain discrete orbits, rather than any path. Each orbit corresponds to an energy level. The energy is quantized, meaning it can only take on specific values.

• The closest orbit to the nucleus has the lowest energy.
• Further orbits have higher energies and are larger.
• Electrons can move between orbits by absorbing or emitting energy.

This idea was groundbreaking because it helped explain why atoms emit light at specific wavelengths.

Hydrogen Atom

The hydrogen atom is the simplest atom, consisting of a single proton and a single electron. This simplicity makes it a perfect candidate to study using the Bohr Model.

In the Bohr model of a hydrogen atom, the electron orbits the proton in circular paths. The model helps to approximate the electron's behavior, even though it isn't perfect. However, for a hydrogen atom, it provides quite accurate predictions.

• Since there is only one electron, calculations are more straightforward.
• The energy levels and orbit radii can be specifically and clearly defined.
• Hydrogen spectra serve as a practical illustration of Bohr's theory.

This model provides foundational knowledge for more complex systems and helps understand atomic structure.

Bohr Radius

The Bohr radius is a key concept in understanding the atomic structure as proposed by Niels Bohr. It is the scale of the smallest possible orbit an electron can have in a hydrogen atom.

The Bohr radius is denoted as \(a_0\), with a value of \(5.29 \times 10^{-11}\) meters. This is a fundamental constant that describes the size of the simplest quantum orbit.

• It represents the most probable distance of the electron from the nucleus when the electron is in its lowest energy state.
• All other orbit radii can be calculated by multiplying the Bohr radius by the square of the principal quantum number.
• The Bohr radius helps relate the size of an atom to its energy levels.

Knowing the Bohr radius is essential for understanding electron distributions and other atomic properties.

Principal Quantum Number

The principal quantum number, denoted as \(n\), is an integral part of the Bohr Model. It essentially defines the sizes and energies of the electron orbits.

This number can take on positive integer values (\(n = 1, 2, 3, \ldots\)), with each integer representing a different orbit or energy level.

• \(n = 1\) corresponds to the ground state, the orbit nearest the nucleus with the lowest energy.
• Higher values of \(n\) denote higher energy levels and larger radii orbits.
• The energy associated with each level increases as \(n\) increases.

The principal quantum number is crucial in determining the arrangement of electrons within an atom and is an essential concept in quantum mechanics.