Question

Consider that \(214 \mathrm{~J}\) of work are done on a system, and \(293 \mathrm{~J}\) of heat are extracted from the system. In the sense of the first law of thermodynamics, what are the values (including algebraic signs) of \((a) W,(b) Q\), and \((c) \Delta E_{\text {int }} ?\)

Short answer

The values (including algebraic signs) of W, Q, and \(ΔE_{int}\) are: W = +214 J, Q = -293 J, and \(ΔE_{int} = -507 J).

Step-by-step solution

Identify the knowns

The work done on the system, W = +214 J and the heat extracted from the system, Q = -293 J (negative because it is extracted from)

Apply first law of thermodynamics for W and Q

W is work done on the system which is +214 J and Q is heat extracted from the system, which is -293 J.

Compute for \(ΔE_{int}\)

Using the first law of thermodynamics, \( ΔE_{int} = Q - W\). Substitute the values, \( ΔE_{int} = -293 J - 214 J = -507 J\). So, the change in internal energy is -507 J.

Key concepts

Work-energy principle

The work-energy principle is a foundational concept in physics that relates the work done by all forces acting on a body to the change in its kinetic energy. This principle can be summarized in the equation:
\[ W = \Delta KE \]
where \( W \) represents the work done on the object, and \( \Delta KE \) denotes the change in kinetic energy.
In a thermodynamic context, work (\( W \)) is the energy transferred by a system to its surroundings, or vice versa, due to macroscopic forces acting over a distance. In the given exercise, seeing that the work is positive (\( +214 \) joules), indicates that energy is being input into the system. This increases the system's energy which could result in an increase in internal energy, temperature, or volume depending on whether the system does work or transfers heat as a result.
However, to fully grasp the effect of work on a system's total energy, one must also consider the changes in internal energy or heat exchanged, as they all play interconnected roles in thermodynamic processes.

Internal Energy

Internal energy, often denoted by \( E_{int} \) or \( U \), is the total energy contained within a system due to the kinetic and potential energy of its molecules. It encompasses all forms of energy internal to a system, excluding the kinetic and potential energies that the system has as a whole with respect to an external frame of reference.
In thermodynamics, changes in internal energy (\( \Delta E_{int} \)) are related to heat (\( Q \)), and work (\( W \)) by the first law of thermodynamics which can be written as:
\[ \Delta E_{int} = Q - W \]
This relation shows that any increase in internal energy must come from either adding heat to the system or doing work on it. Conversely, internal energy decreases when the system loses heat or does work on its surroundings. In our exercise, the negative sign of \( \Delta E_{int} \) signifies that the system's internal energy decreased by \( 507 \) joules, indicating that energy has left the system, either by doing work on the environment or releasing heat.

Thermodynamic Processes

Thermodynamic processes involve the transfer of energy to or from a system in ways that influence the system's properties, such as pressure, volume, temperature, and internal energy. There are several types of thermodynamic processes, including isothermal (constant temperature), adiabatic (no heat transfer), isobaric (constant pressure), and isochoric (constant volume) processes. Each has unique implications for the relationship between work, heat, and internal energy.
In our exercise, the system experiences a process in which work is done on the system and heat is extracted from it. Understanding the context of this process (whether it is open, closed, or isolated, and under what constraints the process occurs) is crucial because it dictates the final state of the system's properties and how they are mathematically related. For instance, if the process were adiabatic, all the work done would change the internal energy, as there would be no heat exchange with the surroundings.
By considering these thermodynamic processes alongside the first law of thermodynamics and the work-energy principle, students can gain a comprehensive view of energy transfer within a physical system.