Show that the mass equivalent of the energy released by the complete combustion of \(1 \mathrm{~mol}\) of methane \((890 \mathrm{~kJ})\) is \(9.89 \mathrm{ng}\).

Einstein's famous equation, \( E = mc^2 \), is one that revolutionized physics. It tells us that energy \( (E) \) and mass \( (m) \) are interchangeable; they are different forms of the same thing. The equation shows that a small amount of mass can be converted into a huge amount of energy, with the conversion factor being the square of the speed of light \( (c^2) \). This speed is an enormous number, \((3.00 \times 10^8 \text{ m/s})^2\), which means that even a tiny amount of mass can be converted into a very large quantity of energy. It’s important to grasp this concept because it underscores not just scientific theory, but also practical applications like nuclear energy generation and our understanding of stellar processes.

The combustion of methane \( (CH_4) \) is a chemical reaction where methane reacts with oxygen to produce carbon dioxide, water, and energy in the form of heat. Methane, being the simplest hydrocarbon, serves as a staple fuel in industries and households.

The Equation for Methane Combustion

The balanced chemical equation is: \( CH_4 + 2 O_2 \rightarrow CO_2 + 2 H_2O + \text{Energy} \). This reaction releases a significant amount of energy, which can be measured in joules (J) or kilojoules (kJ). For our application, knowing that \(1 \text{mol}\) of methane releases \(890 \text{kJ}\) of energy is essential, as it allows us to utilize Einstein’s equation to find the equivalent amount of mass that corresponds to this energy.

The mole concept is a fundamental principle in chemistry that relates the number of particles in a substance to its mass in grams. One mole is defined as the number of atoms in exactly 12 grams of carbon-12, which is approximately \(6.022 \times 10^{23}\) particles, known as Avogadro's number. This concept allows chemists to count particles by weighing them and is crucial for calculations involving chemical reactions. In our methane example, the combustion of \(1 \text{mol}\) of methane implies that we are dealing with Avogadro's number of methane molecules, which collectively release \(890 \text{kJ}\) of energy when they combust.

Energy conversion involves changing one form of energy into another. In the context of mass-energy equivalence, the energy released from the combustion of methane (a chemical energy) can be thought of as being converted into its mass equivalent. This is not a conversion process that we observe in everyday chemical reactions, as the amount of mass converted in such reactions is typically quite small, and hence it is often neglected. However, in the framework of Einstein’s equation, this mass can be calculated and has real implications, especially in understanding high-energy physics and astrophysics.

The calculation of mass equivalent from the energy released involves using Einstein’s equation. Given that the energy release from the combustion of methane is \(890 \text{kJ}\), we convert that to joules to use in the equation: \(890 \text{kJ}\) is anything but simple and underscores the profound nature of the relationship between mass and energy.