Problem 104 - A Complete Resource Book in Chemistry for JEE Main

Question

In one type of mutual annihilation of an electron and a positron, three \(\gamma\)-ray photons are produced. If each photon has an energy of \(0.3407 \mathrm{MeV}\), what is the mass of the positron in amu? ( \(1 \mathrm{amu}=931.5 \mathrm{MeV}\) ) (a) \(7.986 \times 10^{-4}\) (b) \(5.486 \times 10^{-4}\) (c) \(16.86 \times 10^{4}\) (d) \(2.243 \times 10^{-4}\)

Step-by-Step Solution

Answer

(b) \(5.486 \times 10^{-4}\)

Step 1: Understand the Problem

The problem describes the annihilation of an electron and positron which produces three \(\gamma\)-ray photons, each with energy \(0.3407\, \text{MeV}\). We need to find the mass of the positron, knowing \(1\, \text{amu} = 931.5\, \text{MeV}\).

Step 2: Calculate Total Photon Energy

Since there are three photons produced, and each has an energy of \(0.3407\, \text{MeV}\), the total energy of the photons is given by multiplying the energy of one photon by the number of photons: \(3 \times 0.3407\, \text{MeV} = 1.0221\, \text{MeV}\).

Step 3: Relate Energy to Mass

According to Einstein's equation \(E=mc^2\), the energy produced (1.0221 MeV) is equivalent to the mass of the positron-electron pair. Therefore, the mass of the positron should be half of the total energy produced, or \(\frac{1.0221}{2} = 0.51105\, \text{MeV}\).

Step 4: Convert Energy to amu

To find the mass in atomic mass units (amu), we use the conversion factor \(1\, \text{amu} = 931.5\, \text{MeV}\). Thus, the mass in amu is \(\frac{0.51105}{931.5}\, \text{amu} = 5.486 \times 10^{-4}\, \text{amu}\).

Step 5: Select the Answer

Comparing with the given options, (b) \(5.486 \times 10^{-4}\) matches our result.

Key Concepts

Positron-Electron Annihilation

Positron-electron annihilation is a process where a positron (the antimatter equivalent of the electron) and an electron encounter each other. When they meet, they both "annihilate" and transform their combined mass into energy, as predicted by Einstein's famous equation, \( E=mc^2 \). This energy is typically released in the form of photons, specifically gamma-ray photons.

• This is a fundamental concept in particle physics and reflects the mass-energy equivalence principle, where mass can be converted into energy and vice versa.
• The annihilation process conserves energy, charge, and other quantum numbers, highlighting the interconnected principles of physics governing subatomic particles.
Typically, in an annihilation event involving one positron and one electron, two gamma-ray photons are produced. However, variations can result in the production of three or more photons, like in the exercise scenario, demonstrating the principles of conservation in different forms.

Gamma-Ray Photons

Gamma-ray photons are a type of electromagnetic radiation with very high energy. They have the smallest wavelengths and the highest frequencies of all photon types.

• Gamma rays are emitted during radioactive decay or other nuclear and subatomic processes, such as positron-electron annihilation.
• They are capable of penetrating most materials, which is why they are often used in medical imaging and treatments, as well as in industrial applications.

In the annihilation process from the exercise, each of the three gamma-ray photons carries an energy of 0.3407 MeV. This energy is a direct conversion from the mass of the particles involved, showcasing the mass-energy relationship.

Atomic Mass Units (amu)

Atomic mass unit (amu) is a standard unit of mass that quantifies the mass of atoms and subatomic particles. One amu is defined precisely as one twelfth the mass of a carbon-12 atom.

• In energy terms, 1 amu is equivalent to 931.5 MeV, which is a conversion factor from mass to energy, crucial for calculations involving subatomic particles.
• The use of amu simplifies the expression of particle masses and can easily be interchanged with energy units in the matter of conversion.

In the context of the exercise, the mass of a positron was determined by using the conversion from MeV to amu, showing the application of this unit in particle physics calculations and emphasizing the unity of mass and energy at the atomic and subatomic scales. This conversion helps translate complex interactions into more comprehensible terms.