Question

How much matter needs to be converted to energy to supply \(400 \mathrm{~kJ}\) of energy, the approximate energy of \(1 \mathrm{~mol}\) of \(\mathrm{C}-\mathrm{H}\) bonds? What conclusion does this suggest about energy changes of chemical reactions?

Short answer

Only about \( 4.44 \times 10^{-12} \text{ kg} \) of matter needs to be converted. Chemical reactions involve much lower energy changes compared to matter-energy conversion.

Step-by-step solution

Understanding the Energy-Mass Conversion

To find out how much matter needs to be converted into energy, we use the equation from Einstein's theory of relativity, which is given by:\[ E = mc^2 \]where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light in a vacuum (approximately \( 3 \times 10^8 \) meters per second). Given \( E = 400 \) kJ, we need to find \( m \).

Convert Kilojoules to Joules

Energy is provided in kilojoules (kJ) but needs to be in joules (J) for the equation. To convert 400 kJ to J:\[ 400 \text{ kJ} = 400 \times 1000 \text{ J} = 400,000 \text{ J} \]

Solve for Mass Using Einstein's Equation

Rearrange Einstein's equation to solve for mass:\[ m = \frac{E}{c^2} \]Plug in the values:\[ m = \frac{400,000}{(3 \times 10^8)^2} \]Calculate to find the mass.

Perform the Calculation

Compute the value:\[ m = \frac{400,000}{9 \times 10^{16}} \approx 4.44 \times 10^{-12} \text{ kg} \]

Analyze the Result

The mass that needs to be converted is extremely small, approximately \( 4.44 \times 10^{-12} \) kg. This tiny amount of mass conversion can release the same amount of energy as a chemical reaction involving 1 mole of \( \text{C-H} \) bonds.

Key concepts

Einstein's Theory of Relativity

Einstein's Theory of Relativity is a groundbreaking scientific theory that transformed our understanding of space, time, and energy. One of its most famous components is the equation \( E = mc^2 \). This equation signifies that energy \((E)\) and mass \((m)\) are interchangeable; they are essentially different forms of the same thing.

Here, \( c \) represents the speed of light in a vacuum, which is about \( 3 \times 10^8 \) meters per second. It's a colossal number, and since it's squared in the equation, even a small amount of mass can convert into a significant amount of energy. Therefore, this equation explains why atomic reactions release so much energy compared to chemical reactions.

For instance, to release \( 400 \) kilojoules through mass-energy conversion, you would only need to convert about \( 4.44 \times 10^{-12} \) kilograms of mass into energy. This value is extremely small, showcasing the efficiency of mass-energy conversion predicted by the theory.

Chemical Reactions

Chemical reactions involve the rearrangement of atoms within molecules, leading to the formation of different chemical substances. The energy changes in chemical reactions are relatively moderate compared to nuclear processes.

- When a chemical reaction takes place, bonds between atoms are broken, and new ones are formed. This involves energy exchanges.
- The energy absorbed or released during a chemical reaction is usually measured in kilojoules per mole. - In many reactions, the amount of energy involved can be compared to the energy from converted mass, as calculated using Einstein’s equation, to understand their magnitude.

In many everyday chemical reactions, such as the burning of fossil fuels, the energy changes are noticeable. Yet, when we measure how much mass has to "vanish" to create the equivalent energy, the mass is surprisingly negligible. Thus, the immense energy potential hidden within mass explains why chemical reactions can't match the dynamism of nuclear reactions, where Einstein's theory becomes far more apparent.

C-H Bonds Energy

C-H bonds, or Carbon-Hydrogen bonds, are a common and crucial feature in organic chemistry, especially in hydrocarbons and organic compounds.

- These bonds are typically strong, requiring a fair amount of energy to break.
- A single mole of C-H bonds contains about 400 kilojoules of chemical energy.
- Breaking these bonds involves providing energy to overcome the bond strength, while forming new bonds again releases energy.

In the context of the exercise, the energy equivalent of 1 mole of C-H bonds is compared to the massive potential energy that could be released by converting a minute amount of mass into energy, as described by \( E = mc^2 \). This comparison highlights the energy density embedded within molecular bonds and the even greater potential waiting for release in atomic structures.

Understanding the energy within C-H bonds allows chemists to better manipulate organic reactions, optimize energy releases in chemical processes, and develop useful materials and fuels.