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

Short answer

The change in potential energy when a book is transferred from table to shelf (a) and the heat evolved when a cube of sugar is oxidized to CO₂(g) and H₂O(g) (b) are independent of the path by which a change occurs. The work accomplished in burning a gallon of gasoline (c) is dependent on the path and not independent.

Step-by-step solution

A. Analyzing the change in potential energy

When a book is transferred from table to shelf, the change in potential energy is determined by the difference in heights between the table and the shelf. It is given by the equation ΔPE = m × g × Δh, where m is the mass of the book, g is the gravitational constant, and Δh is the difference in heights. In this case, the change in potential energy depends only on the initial and final heights and not on the path taken to transfer the book. Therefore, this process is independent of the path.

B. Analyzing the heat evolved during oxidation of sugar

When a cube of sugar is oxidized to CO₂(g) and H₂O(g), the heat evolved in this process, known as enthalpy change (ΔH), is a state function that depends only on the initial and final states of the system. It is independent of the path or mechanism by which the reaction occurs. Therefore, the heat evolved during the oxidation of sugar is independent of the path.

C. Analyzing the work accomplished in burning a gallon of gasoline

When a gallon of gasoline is burned, the work accomplished depends on various factors such as engine efficiency, driving conditions, and various other factors. The work accomplished is not solely dependent on the initial and final states of the gasoline, but also on how efficiently the energy in the gasoline is utilized during the process. Therefore, the work accomplished in burning a gallon of gasoline is dependent on the path and not independent. Conclusion:

Answer

Among the given processes, the change in potential energy when a book is transferred from table to shelf (a) and the heat evolved when a cube of sugar is oxidized to CO₂(g) and H₂O(g) (b) are independent of the path by which a change occurs. The work accomplished in burning a gallon of gasoline (c) is not independent of the path.

Key concepts

Potential Energy

In physics, potential energy is the energy that an object possesses because of its position in a force field or due to its configuration. An excellent example to understand potential energy is by considering a book being moved from a table to a shelf. The potential energy change, denoted as \( \Delta PE \), is given by the formula: \( \Delta PE = m \times g \times \Delta h \). Here, \( m \) represents the mass of the object, \( g \) is the acceleration due to gravity (typically \( 9.8 \ m/s^2 \) on Earth), and \( \Delta h \) is the change in height.

This equation illustrates that the potential energy only depends on the initial and final heights—not on how the object got there. It doesn't matter whether you lift the book straight up, at an angle, or in zigzag patterns. As long as the starting and ending points are the same, the potential energy change is the same. This independence from the path makes potential energy a state function.

Enthalpy Change

The enthalpy change, denoted as \( \Delta H \), refers to the heat absorbed or released in a chemical reaction at constant pressure. This quantity is crucial in understanding how energy is conserved and transformed during chemical reactions. A good example is the oxidation of sugar, which releases energy in the form of heat that can be measured as an enthalpy change.

Enthalpy is a state function because its value depends only on the system's initial and final states—not how it gets there. In the sugar oxidation process, whether you burn the sugar quickly with a match or let it decompose slowly over time, the total enthalpy change remains the same as long as the initial and final chemical states are identical. This is why we can standardize and tabulate enthalpy changes for various reactions, making them immensely useful in chemical thermodynamics.

State Functions

State functions are properties of a system that depend solely on the current state of the system, not the path taken to reach that state. This means that state functions are determined only by the specific condition of a system at any given time, such as temperature, pressure, and volume, among others.

Common examples of state functions include potential energy, enthalpy, entropy, and internal energy. These properties are intrinsic to the system's current state and allow scientists to use them to predict system changes precisely. For example, when discussing potential energy and enthalpy change, both are state functions since they rely solely on initial and final states rather than the route taken to achieve these states. Understanding state functions helps in formulating laws of thermodynamics and simplifying complex thermodynamic calculations, letting scientists focus on end results rather than intricate processes.