Problem 30 - Chemistry The Central Science In Si Units

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, \((\mathbf{b})\) the heat evolved when a cube of sugar is oxidized to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(g),(\mathbf{c})\) the work accomplished in burning a gallon of gasoline.

Step-by-Step Solution

Answer

Change in potential energy is path-independent.

Step 1: Understanding path independence

In thermodynamics and physics, a path-independent process is also known as a state function. This means the value of the process depends only on the initial and final states, not on the path taken.

Step 2: Assessing Potential Energy Change

The change in potential energy when a book is transferred from a table to a shelf is independent of the path because potential energy is a state function. It only depends on the initial and final heights of the book.

Step 3: Evaluating Heat Evolution

The heat evolved when a cube of sugar is oxidized to produce \(\mathrm{CO}_2(g)\) and \(\mathrm{H}_2\mathrm{O}(g)\) depends on the path taken due to heat being a path function in thermodynamic contexts and closely associated with the process's path.

Step 4: Analyzing Work Accomplished

The work accomplished in burning a gallon of gasoline depends on the path because work is also a path function. Work depends on the specific process by which energy is transferred.

Step 5: Conclusion: Identifying the Path-Independent Scenario

Only the change in potential energy when a book is transferred from a table to a shelf is path-independent, as it is the only state function of the three scenarios.

Key Concepts

Potential Energy

Potential energy is a fascinating concept in physics and thermodynamics. It refers to the energy that an object possesses due to its position or configuration. Imagine a book resting on a table. It has a certain potential energy because of its height above the ground.

• This type of energy relies solely on the object's position.
• Potential energy increases if the object is raised to a higher position, like placing the book on a shelf.

What makes potential energy unique is that it is a state function. This means that its value is determined solely by the initial and final states of the object. Therefore, when moving a book from the table to a higher shelf, it doesn't matter how the book gets there. The change in potential energy only depends on the initial and final heights. In summary, potential energy showcases how energy can be stored due to an object's position. Since it's path-independent, the route taken during the movement does not influence the energy change, making it an essential concept in understanding state functions.

Path Independence

In the world of thermodynamics, understanding what makes a process path-independent is crucial. A process is path-independent if it is determined by the initial and final conditions, rather than the journey or specific steps taken between these states. This concept is often referred to as being a "state function."

• State functions include properties like temperature, pressure, volume, and potential energy.
• These functions are intrinsic to the state of the system, unaffected by how that state was reached.

Potential energy is an excellent example of a path-independent process. When a book moves from a table to a shelf, only the starting and ending positions affect the potential energy change. On the other hand, some processes depend on the path. Heat and work are not path-independent because the amount of energy transferred as heat or work can vary depending on the specific route taken during a process. By focusing on path independence, students can better understand which scenarios can be analyzed using state functions and which require more detailed process-specific information.

Thermodynamics Concepts

Thermodynamics is a branch of physical science concerned with heat and temperature and their relation to energy and work. It involves several crucial concepts, including energy interactions and path dependence.

• Energy interactions in thermodynamics can occur as heat or work.
• State functions, like potential energy, are vital for analyzing equilibrium and changes in state.

Within thermodynamics, path-independent processes play a significant role. These processes help us understand how systems reach equilibrium without needing detailed knowledge of every step. When analyzing thermodynamic processes, it's important to recognize path-dependent quantities, like heat evolved during a chemical reaction, which relies on specific reaction mechanisms. Similarly, work done by a system also varies based on the process's specific path. Equipped with these concepts, students can better analyze and predict the outcomes of various thermodynamic scenarios, distinguishing between state functions and path functions with ease.