Question

The ratio of the radius of the first Bohr orbit for the electron orbiting around the hydrogen nucleus that of the electron orbiting around the deuterium nucleus is approximately (1) \(1: 1\) (2) \(1: 2\) (3) \(2: 1\) (4) \(1: 4\)

Short answer

1:1

Step-by-step solution

Identify the Formula for Bohr Radius

The Bohr radius is given by the formula: \[ r_n = \frac{n^2 \times h^2}{4 \times \frac{\text{\textpi}^2}{e^2} \times m \times Z \times k_e} \] where \( n \) is the orbit number, \( h \) is the Planck's constant, \( e \) is the electric charge, \( m \) is the electron mass, \( Z \) is the atomic number, and \( k_e \) is the Coulomb constant.

Determine the Atomic Numbers for Hydrogen and Deuterium

For hydrogen \( Z = 1 \) and for deuterium (which is an isotope of hydrogen with only one proton, hence same \( Z \)), \( Z = 1 \). Therefore, the atomic number \( Z \) remains 1 for both nuclei.

Simplify the Bohr Radius Formula for Both Nuclei

Since the atomic number \( Z \) and the mass of the orbiting electron \( m \) remain constant for both hydrogen and deuterium, the formula for Bohr radius \( r_n \) remains the same: \[ r_n = \frac{n^2 \times h^2}{4 \times \frac{\text{\textpi}^2}{e^2} \times m \times Z \times k_e} \].

Compare the Radii

Given that no other parameters change and \( Z = 1 \) for both hydrogen and deuterium, the Bohr radius stays the same for an electron orbiting around both types of nuclei. Therefore, the ratio of the radii is \( 1:1 \).

Key concepts

Bohr model

The Bohr model is a crucial concept for understanding atomic structure. Developed by Niels Bohr in 1913, the model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits. This model only works well for simple atoms like hydrogen. According to the Bohr model, each electron orbit, or energy level, has a fixed radius and energy. The most significant formula from the Bohr model is the expression for the radius of these orbits, known as the Bohr radius. These radii are quantized, meaning electrons can only occupy certain orbits and not others.

This model helps explain why electrons do not spiral into the nucleus due to electrostatic attraction. Instead, they remain in stable orbits as long as they do not gain or lose energy. While the Bohr model has limitations, it laid the groundwork for the more sophisticated quantum mechanical models of the atom. Today, it remains a fundamental teaching tool in introductory physics and chemistry courses.

atomic structure

Atomic structure refers to the arrangement of subatomic particles within an atom. Atoms consist of a nucleus containing protons and neutrons, with electrons orbiting this central nucleus. The number of protons (known as the atomic number, Z) determines the chemical element's identity. For example, hydrogen has one proton, and helium has two.

Electrons occupy various energy levels or shells, which can be described through models like the Bohr model. Each energy level can hold a certain number of electrons and higher energy levels are further from the nucleus. The distribution of electrons among these levels determines the atom's chemical properties and reactivity.

The interplay between the subatomic particles—electrons, protons, and neutrons—defines everything from the size of the atom to how it bonds with other atoms. For instance, a hydrogen atom (H) has one proton and one electron, giving it a simple structure. Understanding atomic structure is fundamental to fields such as chemistry, physics, and materials science.

hydrogen isotope

Hydrogen isotopes are variants of the hydrogen atom that have the same number of protons but a different number of neutrons. The most common hydrogen isotope is protium, with one proton and no neutrons. Deuterium, another hydrogen isotope, has one proton and one neutron. Tritium, yet another isotope, has one proton and two neutrons.

Despite the differences in neutron number, all hydrogen isotopes have an atomic number (Z) of 1 because they each have one proton. This means they have the same chemical properties but different physical properties. For example, deuterium, also known as heavy hydrogen, has twice the mass of regular hydrogen (protium) due to the additional neutron.

In the context of the Bohr model, an electron in either a hydrogen or deuterium atom will have very similar orbital radii because the atomic number remains the same. However, properties like the mass of the nucleus will differ. When comparing the Bohr radii of hydrogen and deuterium, we find they are approximately equal due to the identical atomic number, leading to the conclusion that the radius ratio is 1:1.