Problem 66 - A Complete Resource Book in Chemistry for JEE Main

Question

Energy equivalent of \(0.001 \mathrm{mg}\) is (a) \(9 \times 10^{7} \mathrm{ergs}\) (b) \(9 \times 10^{4}\) ergs (c) \(9 \times 10^{7} \mathrm{~J}\) (d) \(9 \times 10^{5} \mathrm{~J}\)

Step-by-Step Solution

Answer

The energy equivalent of 0.001 mg is \(9 \times 10^{7} \text{ J}\), option (c).

Step 1: Understanding the Problem

We must find the energy equivalent of a given mass using Einstein's mass-energy equivalence principle, described by the formula \(E = mc^2\). This requires converting the mass from milligrams to kilograms and applying the speed of light.

Step 2: Convert Mass from Milligrams to Kilograms

Given mass is \(0.001 \text{ mg}\). Convert this to kilograms:\[0.001 \text{ mg} = 0.001 \times 10^{-3} \text{ g} = 0.001 \times 10^{-6} \text{ kg} = 10^{-9} \text{ kg}\]

Step 3: Calculate the Energy using \(E = mc^2\)

The formula \(E = mc^2\) is used to calculate the energy equivalent, where \(m = 10^{-9} \text{ kg}\) and \(c = 3 \times 10^8 \text{ m/s}\), the speed of light. Substitute into the formula:\[E = (10^{-9} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2\]Calculating \((3 \times 10^8 \text{ m/s})^2 = 9 \times 10^{16} \text{ m}^2/\text{s}^2\):\[E = 10^{-9} \times 9 \times 10^{16} = 9 \times 10^{7} \text{ J}\]

Step 4: Conclusion

We derived \(E = 9 \times 10^{7} \text{ J}\), thus the energy equivalent of \(0.001 \text{ mg}\) is \(9 \times 10^{7} \text{ J}\). In ergs, knowing 1 Joule equals \(10^7\) ergs, this remains \(9 \times 10^{7} \text{ J}\).

Key Concepts

Energy Conversion

Energy conversion refers to the process of changing one form of energy into another. In natural processes, energy often transforms rather than being created or destroyed. One of the most important and groundbreaking formulations that showcase energy transformation is found in Einstein's mass-energy equivalence principle. The formula states that a small amount of mass can be converted into a significant amount of energy. By applying the famous equation \(E = mc^2\), we bridge the concept of mass acting as a form of energy storage. Such an understanding allows us to calculate the energy potential within a given mass. For example, converting \(0.001 \, \mathrm{mg}\) of mass into energy demonstrates the enormous energy that can be released from a seemingly minuscule amount of mass. This conversion is pivotal in nuclear reactions, where the mass of nuclear particles is converted into vast amounts of energy.

Mass Conversion

Mass conversion is a significant step in applying the mass-energy equivalence principle. When converting mass for calculations, it's vital to ensure all units are in agreement, particularly using kilogram-meter-second (KMS) units in scientific calculations. In the given exercise, the mass of \(0.001 \, \mathrm{mg}\) is converted into kilograms. This conversion is necessary because the standard unit of mass used in the formula \(E = mc^2\) is kilograms. Here's how it's done:

• First, convert milligrams to grams: \(0.001 \, \mathrm{mg} = 0.001 \times 10^{-3} \, \mathrm{g}\).
• Then, convert grams to kilograms: \(0.001 \times 10^{-3} = 0.001 \times 10^{-6} \, \mathrm{kg}\).
• This results in \(10^{-9} \, \mathrm{kg}\), which is then used in the calculation for energy conversion.

Performing mass conversions correctly ensures accuracy in determining the energy potential according to Einstein's formula. This part of the exercise underlines the importance of unit conversions in scientific calculations.

E = mc^2 Formula

The formula \(E = mc^2\) is one of the most profound equations in physics. Introduced by Albert Einstein, it reveals the relationship between mass (\(m\)) and energy (\(E\)), with \(c\) representing the speed of light in a vacuum. This equation explains that a small amount of mass can produce an enormous amount of energy due to the multiplication factor of the speed of light squared, \(c^2\). Here are the fundamental components:

• \(E\): Energy measured in Joules or ergs.
• \(m\): Mass measured in kilograms.
• \(c\): The speed of light, which is approximately \(3 \times 10^8 \, \text{m/s}\).

By substituting the numbers into the formula in our exercise, we demonstrated that even a mass as small as \(10^{-9} \, \mathrm{kg}\) can be converted into approximately \(9 \times 10^7 \) Joules. This quintessential equation underscores the interconnectedness of mass and energy, leading to breakthroughs in modern physics, including nuclear energy and cosmology.

Speed of Light Calculation

The speed of light, denoted as \(c\), is a key constant in the formula \(E = mc^2\). It is the speed at which light travels in vacuum and is approximately \(3 \times 10^8 \, \text{m/s}\). In our calculations of energy equivalence, the speed of light is squared, \(c^2\), resulting in a massive factor of \(9 \times 10^{16} \, \text{m}^2/\text{s}^2\). This step is crucial because it illustrates why even the conversion of a tiny mass gives rise to such a large amount of energy. Here's why this is important:

• Universal Constant: The speed of light is a universal constant, fundamental to the structure of space-time in the theory of relativity.
• Massive Conversion Factor: Squaring \(c\) highlights the efficiency of mass converting into energy.
• Simplified Calculations: Having a constant allows for consistent and simplified calculations when applied in various physical scenarios.

Thus, understanding and utilizing the speed of light in these calculations are essential to grasp the vast energy capacity stored within matter.