Problem 30 - Chemistry: The Central Science

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 \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), (c) the work accomplished in burning a gallon of gasoline.

Step-by-Step Solution

Answer

Processes (a) and (b) are independent of the path by which a change occurs. The change in potential energy when a book is transferred from a table to a shelf is constant, as it depends only on the height difference. The heat evolved when a cube of sugar is oxidized to \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) also remains constant, as it is determined by the difference in enthalpy, a state function. Process (c), the work accomplished in burning a gallon of gasoline, is not independent of the path, as it depends on various factors such as engine efficiency and energy transfer processes.

Step 1: Analyze Process (a)

We need to determine if the change in potential energy when a book is transferred from a table to a shelf is independent of the path. The potential energy is given by the formula \(PE = mgh\), where \(m\) is the mass of the book, \(g\) is the acceleration due to gravity (approximately \(9.8\,\text{m}\,\text{s}^{-2}\)), and \(h\) is the height difference between the table and the shelf. Here, the height difference remains the same, no matter the path taken. Hence, the potential energy also remains constant, and this process is independent of the path.

Step 2: Analyze Process (b)

Now we analyze the heat evolved when a cube of sugar is oxidized to \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\). The heat evolved in a chemical reaction is determined by the difference in the enthalpy of the products and the reactants. It is a state function, which means that it only depends on the initial and final states, not the path taken for the reaction to occur. So, the heat evolved in this process is also independent of the path.

Step 3: Analyze Process (c)

Lastly, we check if the work accomplished in burning a gallon of gasoline is independent of the path. The work done in burning a gallon of gasoline depends on the efficiency of the engine, the energy transfer process in the system, and other factors that can vary depending on the path. Therefore, the work accomplished in burning a gallon of gasoline is not independent of the path.

Step 4: Conclusion

Based on our analysis, processes (a) the change in potential energy when a book is transferred from a table to a shelf, and (b) the heat evolved when a cube of sugar is oxidized to \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) are independent of the path by which a change occurs, whereas process (c) the work accomplished in burning a gallon of gasoline is not independent of the path.

Key Concepts

Potential Energy

Potential energy is a fundamental concept in physics. It refers to the stored energy an object has due to its position within a force field, most commonly a gravitational field. Consider a book sitting on a table. It possesses potential energy related to the gravitational pull of the Earth.

This potential energy is given by the formula:

• \( PE = mgh \)

where \( m \) represents the mass of the object, \( g \) is the acceleration due to gravity (approximately \( 9.8 \,\text{m/s}^2 \)), and \( h \) is the height above a reference point. The key takeaway from the formula is its dependency solely on height. This is why potential energy is said to be independent of the path taken. Whether you lift the book in a straight line or in a zigzag manner, as long as the height changes by the same amount, the potential energy change remains constant.

Recognizing that potential energy depends only on the initial and final positions can be quite useful. It makes calculations simpler and allows us to predict energy storage in systems efficiently.

State Function

In thermodynamics, a state function is a property whose value does not depend on the path taken to reach that specific value. Rather, state functions are determined solely by the current state of the system. An easy way to grasp this is by comparing it to potential energy – if only the initial and final states matter, then you are dealing with a state function.

One classic example of a state function is enthalpy, often symbolized by \( H \). When a cube of sugar is oxidized to \( \mathrm{CO}_{2}(\mathrm{g}) \) and \( \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \), the heat evolved depends on the difference in enthalpy between the reactants and products. This means that despite the complexity of the reaction steps, the net heat release only depends on the initial and final conditions.

Other examples of state functions include:

• Internal energy
• Pressure
• Volume

Knowing which thermodynamic properties are state functions is valuable because it allows for easier calculations. You don’t need to worry about how a system changed, only where it started and ended.

Work and Energy Concepts

Work and energy are pivotal concepts in the realm of physics and thermodynamics. When discussing work, we are often referring to a force causing an object to move, which may be path-dependent. This means that the amount of work required to move an object can vary based on how the object is moved.

Let's take the example of burning a gallon of gasoline. The work related to this process depends on several factors, like the path or method used in the engine's combustion cycle. In such cases, efficiency, friction, and energy losses all come into play, leading to different work amounts being calculated for the same initial volume of gasoline.
Energy transfer through work is different from energy transfer through heat in that work is not a state function. It involves the exact path taken, making it path-dependent in contrast to state functions like potential energy and enthalpy.

A few pointers to remember:

• Work involves force and movement.
• Work and energy conservation are central to efficiency in physical and chemical processes.
• Always consider path dependency when calculating work.

Recognizing when work is path-dependent can help in designing systems that maximize efficiency and minimize loss.