Question

Indicate which of the following is independent of the path by which a change occurs: (a) the change in potential energy when a book is transferred from table to shelf, (b) the heat evolved when a cube of sugar is oxidized to \(\operatorname{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(g),(\mathbf{c})\) the work accomplished in burning a gallon of gasoline.

Short answer

The process independent of the path by which the change occurs is (a) the change in potential energy when a book is transferred from table to shelf, as potential energy is a state function that depends only on initial and final positions.

Step-by-step solution

Answer

The process that is independent of the path by which the change occurs is (a) the change in potential energy when a book is transferred from table to shelf. This is because the potential energy depends only on the initial and final positions of the book, not on the specific path taken to transfer it. The other processes, (b) and (c), involve heat and work, which are path functions and depend on the steps taken to achieve the change.

Key concepts

Potential Energy

Potential energy is an intriguing concept because it gives us insight into how energy is stored in an object. Imagine lifting a book from a table to a shelf. When you do this, you're increasing the book's potential energy. This increase is due to its position relative to the Earth's gravitational field.

The fascinating part is that potential energy is path independent. This means the energy change doesn't depend on how you move the book, but only on the starting and ending positions. It doesn't matter if you lift it straight up, move it diagonally, or take a loop-the-loop path; the potential energy change remains consistent.

To calculate this energy change, you use the formula for gravitational potential energy: \[ \Delta U = m \cdot g \cdot \Delta h \]where:

• \( m \) is the mass of the book,
• \( g \) represents the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^{2} \)), and
• \( \Delta h \) is the change in height.

So next time you lift something, remember that you're doing more than just moving an object; you're changing its potential energy!

Oxidation

Oxidation is a term often associated with burning or rusting, but it refers more broadly to a chemical reaction. During oxidation, a substance loses electrons, typically interacting with oxygen.

A great example of oxidation is when sugar is burned or metabolized to form carbon dioxide and water. This chemical reaction releases energy, often in the form of heat, which can then be used for various applications. This process provides essential energy in living organisms and is used to generate power through combustion.

Importantly, the heat released during oxidation is path dependent. This means it depends on the specific conditions and steps taken during the reaction. Different methods of oxidizing a substance can result in different amounts of heat.

In metabolism, for instance, sugar oxidation in cells happens via a complex pathway involving many steps to precisely control energy release. So, whether you're digesting a meal or watching wood burn, remember that oxidation is a path followed by chemistry to release energy.

Heat and Work

Heat and work both play crucial roles in thermodynamics, yet they are distinctly different. Heat refers to energy transfer due to temperature differences, while work involves energy transfer when a force moves an object.

Both heat and work are path functions, meaning they depend on the specific path taken between the initial and final states, unlike state functions like potential energy that depend only on these states.

In the case of burning gasoline, the work done depends on the process path. For instance, if you were to burn a gallon of gasoline in a car engine, the work output is determined by the engine cycle's efficiency. Alternative paths or processes might yield different levels of work for the same amount of fuel.

Understanding the path dependency of heat and work helps us optimize processes, like improving engine design for better fuel efficiency. In every energy conversion process, the details matter significantly, influencing how much useful work we can extract or how much heat is involved.