Question

If electron in ground state of \(\mathrm{H}\) -atom is assumed in rest then dipole moment of electron proton system of \(\mathrm{H}\) -atom is \(\ldots \ldots\) Orbit radius of \(\mathrm{H}\) atom in ground state is \(0.56 \AA\). (A) \(0.253 \times 10^{-29} \mathrm{~m}\) (B) \(0.848 \times 10^{-29} \mathrm{~m}\) (C) \(0.305 \times 10^{-29} \mathrm{~m}\) (D) \(1.205 \times 10^{-28} \mathrm{~m}\)

Short answer

The dipole moment of the electron-proton system in the hydrogen atom is approximately \(0.848 \times 10^{-29} \mathrm{~m}\) (Option B).

Step-by-step solution

Identify given information

We are given the orbit radius of a hydrogen atom in the ground state: Orbit radius = 0.56 Å = 0.56 × 10^{-10} meters

Find the charge of an electron and proton

The charge of an electron and proton is equal in magnitude but opposite in sign. For our calculation, we use the magnitude of the charge: Charge of electron (or proton) = 1.6 × 10^{-19} Coulombs

Calculate the dipole moment

Now we can use the formula for dipole moment: dipole moment = charge × distance Dipole moment = (1.6 × 10^{-19} C) × (0.56 × 10^{-10} m)

Evaluate the answer

Evaluate the numerical expression to find the dipole moment of the electron-proton system in the hydrogen atom. Dipole moment = \( (1.6 \times 10^{-19})(0.56 \times 10^{-10}) \) = 0.896 × 10^{-29} Cm Now we compare this result to the options given in the exercise: (A) 0.253 × 10^{-29} m (B) 0.848 × 10^{-29} m (C) 0.305 × 10^{-29} m (D) 1.205 × 10^{-28} m The closest option to our calculated value is (B) 0.848 × 10^{-29} m, which would be the dipole moment of the electron-proton system in the hydrogen atom.

Key concepts

Electron-proton system

The electron-proton system is fundamental to understanding the hydrogen atom's structure. In this system, the tiny negatively charged electron orbits the positively charged proton at the nucleus. These two particles are bound by the electromagnetic force due to their opposite charges. This interaction allows us to calculate various properties like the dipole moment, which is crucial for many physics applications.

The hydrogen atom is the simplest atom, having only one electron-proton pair. The electron's motion around the proton exhibits quantum mechanics properties, such as quantized energy levels. However, for this exercise, we're assuming the electron is at rest, making it easier to calculate the dipole moment.

The system's dipole moment is a measurement of the separation of charges in a molecule. To find the dipole moment, we consider the strength of the charge and the distance between the electron and proton. This helps us to derive insights into the atom's behavior and interactions.

Orbit radius in hydrogen atom

The orbit radius in a hydrogen atom is a key aspect when studying atomic structures. In the exercise, we consider the orbit radius in the hydrogen atom's ground state, which is the lowest energy level it can occupy.

The given orbit radius for the hydrogen atom is 0.56 Å, which converts to 0.56 × 10^{-10} meters. This radius represents the average distance between the electron and proton in the simplest possible orbit.

Understanding the orbit radius helps in identifying the atom's spatial characteristics and its behavior in various states. It also plays a role in calculating the dipole moment, where the distance between the electron and proton is a crucial factor. In essence, knowing the orbit radius allows us to delve deeper into atomic interactions and configurations.

Elementary charge

The elementary charge is a basic property of particles like electrons and protons. It signifies the smallest unit of charge that can be isolated and is denoted as 'e.' The magnitude of this charge is about 1.6 × 10^{-19} Coulombs.

Both the electron and the proton have charges equal in magnitude but opposite in sign. The electron carries a negative charge, while the proton carries a positive charge. This equal magnitude is crucial for many calculations involving atomic interactions, like the dipole moment calculation in the electron-proton system.

In the context of the hydrogen atom, knowing the elementary charge helps us in understanding the fundamental forces at play within the atom. Whether calculating electromagnetic interactions or predicting the behavior of atoms in various conditions, the elementary charge serves as a cornerstone to these investigations.