Question

Which of the following is correct equation? (a) \(\Delta \mathrm{U}=\Delta \mathrm{Q}-\mathrm{W}\) (b) \(\Delta \mathrm{W}=\Delta \mathrm{U}+\Delta \mathrm{Q}\) (c) \(\Delta \mathrm{U}=\Delta \mathrm{W}+\Delta \mathrm{Q}\) (d) none of these

Short answer

The correct equation is (a) \( \Delta \mathrm{U} = \Delta \mathrm{Q} - \mathrm{W} \).

Step-by-step solution

Understand the First Law of Thermodynamics

The First Law of Thermodynamics is generally stated as \( \Delta \mathrm{U} = \mathrm{Q} - \mathrm{W} \), where \( \Delta \mathrm{U} \) represents the change in internal energy, \( \mathrm{Q} \) is the heat added to the system, and \( \mathrm{W} \) is the work done by the system. This formula essentially reflects the principle of conservation of energy for a thermodynamic system.

Compare with Given Options

Compare the given options to the standard equation derived from the First Law of Thermodynamics.- (a) \( \Delta \mathrm{U} = \Delta \mathrm{Q} - \mathrm{W} \)- (b) \( \Delta \mathrm{W} = \Delta \mathrm{U} + \Delta \mathrm{Q} \)- (c) \( \Delta \mathrm{U} = \Delta \mathrm{W} + \Delta \mathrm{Q} \)- (d) none of theseClearly, option (a) correctly matches the standard equation format \( \Delta \mathrm{U} = \Delta \mathrm{Q} - \mathrm{W} \).

Verify Correctness of Each Option

- Check (a): It matches \( \Delta \mathrm{U} = \mathrm{Q} - \mathrm{W} \), therefore it's correct.- Check (b): Rearrange it as \( \Delta \mathrm{W} = \Delta \mathrm{U} + \Delta \mathrm{Q} \), which is incorrect based on the First Law.- Check (c): Rearrange it as \( \Delta \mathrm{U} = \Delta \mathrm{W} + \Delta \mathrm{Q} \), which is also incorrect.- Check (d): Ascertained to be false since option (a) is correct.

Key concepts

Internal Energy

Internal energy is a fundamental concept in the field of thermodynamics. It refers to the total energy contained within a thermodynamic system, consisting primarily of the microscopic energy of particles, including potential and kinetic energy. This concept is crucial to understand because changes in internal energy ( U) are central to energy analysis within thermodynamic systems.

When we talk about the transfer of energy, we usually describe how heat enters or leaves the system and how work is performed by or on the system. These changes in heat and work lead to changes in internal energy. In a closed system, the internal energy changes as it either absorbs energy in the form of heat or performs work on its surroundings. This relationship is expressed using the equation from the First Law of Thermodynamics:

• The change in internal energy ( U) is equal to the heat added to the system (Q) minus the work done by the system (W).

In simpler terms, if a system receives heat, its internal energy increases unless the system also does work which reduces internal energy.

Conservation of Energy

The principle of conservation of energy is a core tenet of physics and is elegantly encapsulated in the First Law of Thermodynamics. Essentially, this law states that energy cannot be created or destroyed, but it can be transformed from one form to another. This means that the total energy of an isolated system remains constant over time.

In the context of a thermodynamic process, the First Law can be summarized as follows:

• When energy is transferred to or from a system as heat, or when work is done by or on the system, the system's internal energy changes according to its conservation principles.
• The mathematical expression:  U = Q - W, where  U is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.

This relationship demonstrates that the increase in energy of a system (by adding heat) can be offset by the energy taken away as work. Ultimately, the First Law highlights the balancing act of energy within a thermodynamic system, reinforcing that quantity-wise, energy is conserved.

Thermodynamic System

To grasp the concept of thermodynamics, it's essential to understand what a thermodynamic system is. It refers to a specific part of the universe that is under study. Everything outside this system is considered the environment or surroundings.

Thermodynamic systems come in three main types:

• **Open systems:** These allow the exchange of both energy and matter with their surroundings.
• **Closed systems:** Energy can cross the boundary, but matter cannot.
• **Isolated systems:** Neither energy nor matter can enter or leave.

When analyzing a thermodynamic process, understanding what type of system you are dealing with helps determine the kind of exchanges taking place. For example, a closed system's analysis would focus on how energy in the form of heat and work affects internal energy changes because the matter is not exchanged. Each type of system responds differently under the principles of energy conservation, affecting the internal energy and transformation processes described by the First Law of Thermodynamics.