Problem 18 - Master Resource Book in JEE Main Physics

Question

One amu is equivalent to \(931 \mathrm{MeV}\) energy. The rest mass of electron is \(9.1 \times 10^{-31} \mathrm{~kg}\). The mass energy is (1 amu \(=1.67 \times 10^{-17} \mathrm{~kg}\) ) (a) \(0.5073 \mathrm{MeV}\) (b) \(0.693 \mathrm{MeV}\) (c) \(4.0093 \mathrm{MeV}\) (d) None of these

Step-by-Step Solution

Answer

(a) 0.5073 MeV

Step 1: Calculate Mass of Electron in amu

First, we need to convert the mass of an electron from kilograms to atomic mass units (amu). Given:- Mass of electron, \( m_e = 9.1 \times 10^{-31} \) kg- 1 amu = \( 1.67 \times 10^{-27} \) kgTo find the mass of the electron in amu, we use:\[ \text{Mass of electron in amu} = \frac{m_e}{\text{mass of 1 amu}} = \frac{9.1 \times 10^{-31}}{1.67 \times 10^{-27}} \approx 0.000545 \text{ amu} \]

Step 2: Convert Electron Mass to Energy in MeV

Now, convert the electron's mass from amu to energy in MeV using the fact that 1 amu is equivalent to 931 MeV.\[ \text{Energy} = 0.000545 \text{ amu} \times 931 \text{ MeV/amu} \]Solving for the energy:\[ \text{Energy} \approx 0.5073 \text{ MeV} \]

Step 3: Compare with Given Options

The calculated energy for the rest mass of an electron is approximately 0.5073 MeV. Comparing this with the given options, we see: (a) 0.5073 MeV (This matches our calculated value.) (b) 0.693 MeV (c) 4.0093 MeV (d) None of these Thus, option (a) is the correct answer.

Key Concepts

Atomic Mass Unit (amu)

The atomic mass unit, abbreviated as amu, is a standard unit of mass used to express the masses of atoms and molecules. It is crucial in chemistry and physics for simplifying calculations related to atomic scale processes. One amu is defined based on the carbon-12 isotope, where one atom of carbon-12 is exactly 12 amu.
This makes amu a useful conversion factor connecting atomic mass with real-world mass figures, such as kilograms. For example, in this problem, 1 amu is considered equivalent to around \(1.67 \times 10^{-27}\) kg.

• It allows scientists to convert between kilograms and atomic masses with ease.
• Here, the conversion of the electron's mass \(9.1 \times 10^{-31} \) kg into amu showcases its practicality in particle physics.

Understanding amu is essential when calculating energy and mass at an atomic scale, bridging macroscopic measurements to tiny atomic and subatomic levels.

Electron Rest Mass

The electron rest mass is the mass of an electron when it is not in motion, and it is a foundational concept in particle physics. Electrons are subatomic particles found outside the nucleus of an atom. They are negatively charged and help balance the positively charged protons in the atomic structure.
The rest mass of an electron is \(9.1 \times 10^{-31} \) kg, which seems incredibly small but is substantial from the perspective of atomic particles.
This value is used in equations to predict behaviors in interactions such as electromagnetic processes and mass-energy equivalence.

• Utilizing electron rest mass in calculations allows the accurate determination of mass-related energy.
• Each tiny electron contributes significantly to the energy output due to its rest mass.

Recognizing the contribution of electron mass at rest helps us appreciate the intricate balance required in atoms and its relation to energy.

Energy Conversion in MeV

Energy conversion, particularly from mass to energy, is a profound concept exemplified by the equation \(E=mc^2\) from Einstein’s theory of relativity. In this context, nuclear and particle physics often use the electronvolt (eV) as a unit of energy, with MeV representing a million electronvolts. This unit is convenient for describing the energies typical in atomic and nuclear processes.
In our example, it turns out that 1 amu is equivalent to 931 MeV. This conversion factor allows for the calculation of energy related to atomic mass. Here's how it's applied:

• The electron mass (in amu) \(0.000545\) multiplied by \(931 \text{ MeV/amu}\) yields approximately \(0.5073 \mathrm{MeV}\).
• Converting mass to energy provides insight into the potential energy within a particle's mass.

This conversion is critical for understanding phenomena like nuclear reactions where mass is transformed into energy.