Problem 75 - Master Resource Book in JEE Main Physics

Question

Assertion \(1 \mathrm{amu}\) is equivalent to \(931 \mathrm{MeV}\). Reason Energy equivalent \((E)\) or mass \((m)\) is \(E=m c^{2} .\)

Step-by-Step Solution

Answer

The assertion is correct; 1 amu is approximately equal to 931 MeV due to mass-energy equivalence.

Step 1: Understanding Assertion

The assertion states that 1 amu (atomic mass unit) is equivalent to 931 MeV. We need to verify this statement by calculating the energy equivalent of 1 amu using known physics formulas.

Step 2: Using Mass-Energy Equivalence

The mass-energy equivalence formula is given by Einstein's famous equation: \[ E = mc^2 \] where \(E\) is the energy, \(m\) is the mass, and \(c\) is the speed of light.

Step 3: Calculating Energy for 1 amu

First, convert 1 amu to kilograms, since energy calculations use SI units. 1 amu = \(1.66 \times 10^{-27}\) kg. Using \( c = 3 \times 10^8 \) m/s, we find: \[ E = (1.66 \times 10^{-27} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2 \] \[ E = 1.494 \times 10^{-10} \text{ J} \]

Step 4: Converting Joules to MeV

To compare with the assertion, we need to convert the energy from joules to MeV. Using the conversion factor \(1 \text{ J} = 6.242 \times 10^{12} \text{ MeV}\), we get: \[ E = 1.494 \times 10^{-10} \text{ J} \times 6.242 \times 10^{12} \text{ MeV/J} \approx 931 \text{ MeV} \]

Step 5: Conclusion

The calculation confirms that 1 amu approximately equals 931 MeV. Therefore, the assertion is correct. The reason supports this, as it refers to the mass-energy equivalence formula used in the calculation.

Key Concepts

Atomic Mass Unit

An atomic mass unit, commonly abbreviated as amu, is a standard unit of mass that quantifies the weight of atoms and molecules. It's one of the smallest mass units used in chemistry and physics, and it is defined as one twelfth of the mass of a carbon-12 atom. This is essential because it allows scientists to express atomic and molecular weights with convenient numbers.

When discussing energy calculations, the atomic mass unit is key because it connects the microcosm of atomic particles to practical measurements in physics through conversions like that of energy. Since these particles are incredibly small, amu provides a manageable method to calculate their mass and subsequently their energy, when using physical equations like Einstein's mass-energy equivalence equation.

Energy Conversion

Energy conversion is the process of changing one form of energy into another. In the context of atomic physics, it's especially important when dealing with conversions from mass to energy, as highlighted in the problem we're discussing.

When we say 1 amu is equivalent to 931 MeV, this illustrates the conversion between units of mass (amu) and units of energy (MeV), achieved through precise calculations. Understanding energy conversion helps in comprehending how microscopic mass can generate substantial energy, consistent with nuclear reactions and decay processes.
Moreover, it highlights the interplay of different measurement units used in physics, allowing scientists to accurately interpret and contextualize observations in both theoretical and experimental physics.

Einstein's Equation

Einstein's famous equation, \(E = mc^2\), represents one of the pivotal principles in physics: mass-energy equivalence. Essentially, it states that mass can be converted into energy and vice versa. In this equation, \(E\) denotes energy, \(m\) represents mass, and \(c\) is the speed of light in a vacuum, which is approximately \(3 \times 10^8\) meters per second.

This equation fundamentally altered the understanding of physics by showing that mass was not an immutable property of objects, but rather a form of energy. In our given problem, we've applied this equation to demonstrate how a small mass in atomic mass units can be converted to a large energy value in MeV.

• It unifies the concepts of mass and energy into one, illustrating that they are interchangeable.
• Provides insight into the significant energy yield of nuclear reactions.
• Is critical in the calculations involving atomic and subatomic particles.

Understanding Einstein's equation facilitates a deeper appreciation of the universe's fundamental forces and behaviors.