Question

According to Einstein's equation, the conversion of \(1.00 \mathrm{~g}\) of matter into energy generates how much energy?

Short answer

The energy generated is \( 9.00 \times 10^{13} \) Joules.

Step-by-step solution

Understanding Einstein's Equation

Einstein's mass-energy equivalence is represented by the equation: \[ E = mc^2 \]where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light in a vacuum.

Converting Mass to Kilograms

The mass given is \( 1.00 \mathrm{~g} \). To use the equation \( E = mc^2 \), the mass must be in kilograms. 1 gram is equal to \( 0.001 \) kilograms:\[ 1.00 \, \mathrm{g} = 0.001 \, \mathrm{kg} \]

Using the Speed of Light

The speed of light \( c \) is \( 3.00 \times 10^8 \) meters per second. This value is constant and is used in the equation.

Calculating Energy

Plug the values of \( m = 0.001 \mathrm{~kg} \) and \( c = 3.00 \times 10^8 \mathrm{~m/s} \) into the equation:\[E = (0.001 \, \mathrm{kg})(3.00 \times 10^8 \, \mathrm{m/s})^2\]Calculate the square of the speed of light first:\[(3.00 \times 10^8 \, \mathrm{m/s})^2 = 9.00 \times 10^{16} \, \mathrm{m^2/s^2}\]Then, multiply by the mass:\[E = 0.001 \, \mathrm{kg} \times 9.00 \times 10^{16} \, \mathrm{m^2/s^2} = 9.00 \times 10^{13} \, \mathrm{J}\]Thus, the energy generated is \( 9.00 \times 10^{13} \) Joules.

Key concepts

E=mc^2

The famous equation \( E=mc^2 \) is one of Albert Einstein's most groundbreaking contributions to physics. It stands for energy \( E \), equals mass \( m \), times the speed of light \( c \) squared. This compact equation reveals the powerful relationship between mass and energy.

- **Mass and Energy**: The equation implies that mass can be converted into energy, and vice versa. This has profound implications in both theory and practice.
- **Dimensional Consistency**: In this equation, you always need to ensure that the mass is measured in kilograms to maintain consistency with energy's unit of Joules.

In summary, \( E=mc^2 \) not only encapsulates a deep physical truth but also provided a foundation for modern physics, influencing areas such as nuclear energy and cosmology.

speed of light

The speed of light, denoted by \( c \), is a fundamental constant in physics, profoundly affecting many areas of study. Measured as \( 3.00 \times 10^8 \) meters per second, the speed of light is a key component in Einstein's equation.

- **In Vacuum**: The speed of light is constant in a vacuum, which serves as the universal speed limit for information and matter.
- **Role in Relativity**: Einstein's theory of relativity relies heavily on the constancy of the speed of light, establishing how time and space are interconnected.

The invariability of the speed of light makes it an invaluable tool for calculations in physics, providing the bridge between mass and energy.

energy conversion

Energy conversion is the process of changing energy from one form to another. In the context of \( E=mc^2 \), it particularly refers to transforming mass into energy.

- **Practical Examples**: Nuclear reactions in the sun and nuclear power plants are practical examples where mass is converted into significant amounts of energy.
- **Types of Energy**: The energy produced can take many forms such as kinetic, thermal, or electrical energy.

Energy conversion is a critical concept in energy management and technology, underlying its applications in both natural processes and human-engineered systems.

mass to energy conversion

Mass to energy conversion involves transforming the mass of an object into energy as defined by \( E=mc^2 \).

- **Matter and Antimatter**: When matter and antimatter collide, they annihilate each other, converting their entire mass into energy.
- **Nuclear Fusion and Fission**: These are processes where mass is converted to energy, driving phenomena like the power of the sun and nuclear power plants.

Understanding mass to energy conversion enhances our ability to harness clean and powerful energy sources, demonstrating how the theory finds relevance in practical applications.