Question

The mathematical form of the first law of thermodynamics is (a) \(d q=d U=d w\) (b) \(d q=d U-d w\) (c) \(d U=d q+d w\) (d) none of these

Short answer

(b) \(dq = d U - d w\)

Step-by-step solution

Identify the correct mathematical form

Compare each of the given choices (a), (b), (c), and (d) with the true mathematical form of the first law of thermodynamics. The correct formulation is \(d U = d q - d w\).

Match the correct choice

Upon comparison, it's clear that (b) \(dq = d U - d w\) matches with the correct mathematical form of the first law of thermodynamics.

Key concepts

Thermodynamic equations

Thermodynamic equations form the foundation of our understanding of how energy is transferred and transformed in thermodynamic systems. The First Law of Thermodynamics is one such equation that is crucial in physics and engineering. This law, also known as the principle of energy conservation, essentially states that energy cannot be created or destroyed. It can only change forms. In equation form, it is expressed as:

• \( dU = dQ - dW \)

This equation tells us that the change in internal energy \(dU\) of a system is equal to the heat added to the system \(dQ\), minus the work done by the system \(dW\).
The First Law provides a framework for analyzing energy exchanges and transformations in thermodynamic processes. It applies universally, from simple systems like a gas in a piston to complex systems like living organisms or power plants.

Internal energy

Internal energy is a central concept in thermodynamics, representing the total energy contained within a system. This energy includes kinetic energy due to the movement of particles and potential energy from forces between particles. It is crucial to understand that internal energy is not something we can measure directly.
Instead, we observe changes in internal energy, which we denote as \(dU\). Changes occur when energy is transferred into or out of the system, either by heat or work.

• When heat is added to a system, internal energy increases.
• When work is done by the system, internal energy decreases.
• A balance between heat added and work done determines whether internal energy increases, decreases, or stays constant.

Understanding internal energy is vital for analyzing how energy flows and transformations occur within thermodynamic systems.

Heat transfer

Heat transfer is the process by which thermal energy moves from one body or system to another. It occurs when there is a temperature difference between two entities. Heat naturally flows from hot to cold and can transfer energy across boundaries in various ways.
Some common methods of heat transfer include:

• **Conduction** - Heat flows through a material or substance, such as metal, through direct contact.
• **Convection** - Heat transfer occurs through the movement of fluids, like air or water.
• **Radiation** - Transfer of heat in the form of electromagnetic waves, such as radiant heat from the Sun.

In the context of the First Law of Thermodynamics, heat transfer \(dQ\) is one of the main forms of energy exchange, affecting the internal energy of a system directly.

Work done

Work done \(dW\) in thermodynamics refers to the energy transferred by a system to its surroundings through macroscopic forces. For instance, when a gas expands in a piston-cylinder assembly, it pushes against the piston, doing work on its environment.
This concept is essential as it helps determine how the system interacts with its surroundings. Work can be either:

• **Positive** - If the system does work on the surroundings, like a gas expanding.
• **Negative** - If work is done on the system, such as compressing a gas.

The work done is an integral part of the First Law of Thermodynamics defined in the equation \(dU = dQ - dW\). This shows that the amount of work done by the system influences the change in internal energy. Understanding this helps us predict and measure how energy flows in and out of thermodynamic systems.