Question

In the second footnote on page 42 it was pointed out that mass and energy are alternate aspects of a single entity called mass-energy. The relationship between these two physical quantities is Einstein's equation, \(E=m c^{2}\), where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light. In a combustion experiment, it was found that \(12.096 \mathrm{~g}\) of hydrogen molecules combined with \(96.000 \mathrm{~g}\) of oxygen molecules to form water and released \(1.715 \times 10^{3} \mathrm{~kJ}\) of heat. Use Einstein's equation to calculate the corresponding mass change in this process, and comment on whether or not the law of conservation of mass holds for ordinary chemical processes.

Short answer

The mass change is \(1.906 \times 10^{-11} \text{ kg}\), which is negligible in chemical processes, supporting the conservation of mass.

Step-by-step solution

Understanding Einstein's Equation

Einstein's equation is given by \(E = m c^{2}\), where \(E\) is energy, \(m\) is the mass change, and \(c\) is the speed of light in a vacuum (approximately \(3.00 \times 10^{8} \text{ m/s}\)). This equation shows that energy and mass are interchangeable.

Convert Energy from kJ to Joules

The energy released is given in kilojoules as \(1.715 \times 10^{3} \mathrm{~kJ}\). Convert this energy into joules by using the conversion factor \(1 ext{ kJ} = 1000 ext{ J}\). Thus, the energy in joules is \(1.715 \times 10^{3} imes 10^{3} = 1.715 imes 10^{6} ext{ J}\).

Calculate Mass Change

Substitute the energy \(E = 1.715 \times 10^{6} ext{ J}\) and the speed of light \(c = 3.00 \times 10^{8} ext{ m/s}\) into Einstein's equation. We rearrange the equation to solve for the mass change \(m\): \[m = \frac{E}{c^{2}}\]. Substituting the values gives \[m = \frac{1.715 \times 10^{6}}{(3.00 \times 10^{8})^{2}}\].

Perform Calculation

First, calculate \(c^{2}\): \((3.00 \times 10^{8})^{2} = 9.00 \times 10^{16}\). Then, divide the energy by this value to find the mass change: \[m = \frac{1.715 \times 10^{6}}{9.00 \times 10^{16}} = 1.906 \times 10^{-11} ext{ kg}\].

Evaluate the Conservation of Mass

The calculated mass change, \(1.906 \times 10^{-11} ext{ kg}\), is extremely small. In ordinary chemical processes like combustion, the mass change is negligible compared to the total mass involved, hence the law of conservation of mass is effectively upheld at this scale. In nuclear reactions, where energy changes are more significant, the mass change is noticeable.

Key concepts

Conservation of Mass

The conservation of mass is a fundamental concept in chemistry. It states that mass cannot be created or destroyed in a closed system. During chemical reactions, the mass of the reactants should equal the mass of the products. In other words, the total amount of matter remains constant.

However, in processes involving significant energy changes, small discrepancies can occur, as some mass is converted into energy or vice versa according to mass-energy equivalence. This deviation is usually minute in ordinary chemical reactions, such as the combustion of hydrogen and oxygen to form water, making it practically unnoticeable.

• In typical chemical processes, mass appears conserved because the changes in mass are incredibly small.
• The small mass changes become more noticeable in reactions involving significant energy changes, like in nuclear reactions.

Einstein's Equation

Einstein's equation, expressed as \(E=mc^2\), demonstrates the powerful relationship between mass and energy. Here, \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum, approximately \(3.00 \times 10^8 \text{ m/s}\). This equation implies that mass can be converted into energy and vice-versa.

In practical terms, it means that a small amount of mass can be transformed into a huge amount of energy due to the large value of \(c^2\). This principle is crucial in modern physics, affecting everything from nuclear power to cosmological theories.

• \(c\) is a large number, making \(c^2\) extremely large, which is why even small mass changes can produce substantial energy.
• This equation is essential in understanding phenomena where energy and mass quantities are interchangeable.

Chemical Processes

Chemical processes involve the rearrangement of atoms to form new substances. In these processes, the conservation of mass generally holds true because the changes in mass from energy conversions are typically negligible.

Taking the example of burning hydrogen to produce water, we see hydrogen and oxygen molecules reacting to produce heat and water. While Einstein's equation shows a minuscule mass change due to energy release, it is too tiny to observe without precise equipment.

• Typical chemical reactions assume mass conservation due to negligible mass-energy conversion.
• Nuclear reactions differ as they involve noticeable mass changes due to significant energy emissions.

Energy Conversion

Energy conversion is a process where energy changes from one form to another, like chemical energy converting to heat during combustion. Einstein's equation provides a framework to also include mass transformations in these energy conversions.

In the reaction of hydrogen with oxygen, the energy released as heat comes from a tiny amount of mass reduced according to \(E=mc^2\). Although this change is unnoticeable in chemical reactions, it highlights the unity of mass and energy.

• In ordinary chemical processes, mass-energy conversion is hardly perceptible.
• In high-energy systems, like nuclear reactions, this conversion is significant and crucial.