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.

Short answer

The correct answer is option (a): the change in potential energy when a book is transferred from table to shelf is independent of the path by which a change occurs, as it only depends on the initial and final positions of the book.

Step-by-step solution

Process (a): Change in potential energy of a book transferred from table to shelf

In this process, we are comparing the potential energy of a book when it is placed on a table to its potential energy when it is placed on a shelf. The potential energy difference depends on the change in height between the two locations and the weight of the book. Since this change in potential energy depends only on the initial and final positions of the book and not on the path taken to move it, this process is independent of the path. So, option (a) is the correct answer.

Process (b): Heat evolved when a cube of sugar is oxidized to CO2(g) and H2O(g)

In this process, we are asked to consider the heat evolved during the oxidation of sugar. The heat evolved in this process depends on the chemical reactions that occur during the oxidation process and not on the specific steps taken to initiate or perform these reactions. Therefore, the heat evolved when a cube of sugar is oxidized to CO2(g) and H2O(g) is independent of the path taken, but we already selected option (a) as the correct answer.

Process (c): Work accomplished in burning a gallon of gasoline

In this process, we are asked to consider the work accomplished when a gallon of gasoline is burned. The work accomplished during this process depends on the efficiency of the engine in which the gasoline is burned and the mechanical processes used to transform the chemical energy of the gasoline into useful work. This efficiency can vary depending on the type of engine, its operating conditions, and other factors. Therefore, the work accomplished in burning a gallon of gasoline is dependent on the path taken, so this option is not the correct answer. In conclusion, the correct answer to this question is option (a): the change in potential energy when a book is transferred from table to shelf is independent of the path by which a change occurs.

Key concepts

Potential Energy

Potential energy is a fundamental concept in physics and chemistry, which refers to the energy that is stored within an object due to its position or arrangement. For instance, when a book is placed on a higher shelf, it has more potential energy compared to when it is on a lower table because of the increased height above ground level. The potential energy is calculated using the formula \( PE = m \times g \times h \), where \(m\) is the mass of the object, \(g\) the acceleration due to gravity, and \(h\) the height above the reference point.

Potential energy is an important concept in chemistry, especially when considering molecular structures and the forces between atoms and ions. Chemical potential energy is stored in the bonds between atoms and is released or absorbed during chemical reactions. The change in potential energy is independent of the path taken and depends only on the initial and final states of the system. Understanding this concept helps students see the conservation of energy within a system and its conversion from one form to another without being impacted by the process used to initiate a change.

Heat of Reaction

The heat of reaction, also known as enthalpy change, is the heat absorbed or released during a chemical reaction. In chemistry, the heat of reaction is a critical concept because it helps scientists and engineers to understand how much energy is involved in the making and breaking of chemical bonds. When a sugar cube oxidizes to carbon dioxide (\(CO_2\)) and water vapor (\(H_2O\)), the heat evolved is the difference in the potential energy between the reactants and the products.

The heat of reaction is determined by the properties of the substances involved and the stoichiometry of the reaction—not the steps taken to carry out the reaction. This is a state function, meaning it is path-independent. In educational content, emphasizing that the heat of reaction is a crucial factor in energy production and consumption, and has practical implications in designing sustainable and energy-efficient reactions in processes like manufacturing and transportation, reinforces its relevance.

Work and Energy

In the context of thermodynamics, work and energy are closely related but distinct concepts. While energy encompasses the overall capacity to do work or produce heat, work itself is defined as the force applied to an object over a distance. Work is given by the formula \( W = F \times d \) where \(W\) is work, \(F\) is force, and \(d\) is the distance over which the force is applied.

In thermodynamics, work can be done by a system (such as an engine) or on a system (like compressing a gas). The critical point to note is that work is path-dependent; it reflects the processes that happen between the initial and final states. For instance, the work accomplished by burning gasoline in an engine depends on the engine's design and operating conditions, which can vary the efficiency of how this chemical energy is converted into mechanical work. Clarifying this concept is essential for students learning about energy efficiency and the design of machines and engines in various fields of study.