Problem 56 - Pearson Physics

Question

Which of the physical quantities, \(Q, W\), or \(\Delta E\), is zero in a constant-volume process?

Step-by-Step Solution

Answer

In a constant-volume process, work done (W) is zero.

Step 1: Understanding Constant-Volume Processes

In a constant-volume process, the volume of the system does not change. This means there is no expansion or compression work done by or on the system.

Step 2: Identifying Work Done (W)

Work done by or on a system due to volume change can be expressed as \( W = P imes \Delta V \), where \( P \) is pressure, and \( \Delta V \) is the change in volume. Since the process occurs at constant volume, \( \Delta V = 0 \), leading to \( W = 0 \).

Step 3: Analyzing Heat Transfer (Q) and Change in Internal Energy (\(\Delta E\))

In a constant-volume process, heat transfer \( Q \) changes the internal energy \( \Delta E \) of the system. This is described by the first law of thermodynamics: \( \Delta E = Q - W \). Since \( W = 0 \), it simplifies to \( \Delta E = Q \), indicating that \( \Delta E \) is equal to \( Q \).

Step 4: Determining Zero Quantity

Given that \( W = 0 \) in a constant-volume process, while \( \Delta E \) and \( Q \) are potentially non-zero depending on the heat interaction.

Key Concepts

Work Done in Constant-Volume Processes

In a constant-volume process, the system's volume remains unchanged. This means there is no displacement of any boundaries, so no work is done. Work is typically defined as the energy transferred when an external force is applied to a system, causing it to move or deform.
For thermodynamic systems, work done due to volume change is given by the formula:

• \( W = P \times \Delta V \)

Here, \( P \) is the pressure, and \( \Delta V \) is the change in volume. Since the volume does not vary in this process, \( \Delta V = 0 \), which results in \( W = 0 \). Note that no other type of work interaction compensates since the volume constraint implies that expansion or compression work does not take place.

Internal Energy in Constant-Volume Processes

Internal energy, denoted by \( E \), is the total energy contained within a system. It includes kinetic and potential energy of the particles that comprise the system. Changes in internal energy, \( \Delta E \), occur when energy is added or removed from the system, typically through heat or work interactions.
In the context of a constant-volume process, the absence of work means that any change in internal energy is solely due to heat transfer. That is, if energy is added to the system as heat, the internal energy increases by an equivalent amount. Since work done \( W = 0 \), the first law of thermodynamics simplifies to:

• \( \Delta E = Q \)

Where \( Q \) is the heat added to or removed from the system. This relationship highlights how changes in internal energy reflect the heat flow under such conditions.

First Law of Thermodynamics in Constant-Volume Processes

The first law of thermodynamics is a principle of energy conservation. It states that the change in internal energy of a system equals the heat added to the system minus the work done by the system. Mathematically, this is expressed as:

• \( \Delta E = Q - W \)

In a constant-volume setting, the work term \( W \) becomes zero due to zero volume change. This simplifies the equation to:

• \( \Delta E = Q \)

This expression shows a direct relationship between the heat exchanged \( Q \) and the change in internal energy \( \Delta E \). It underscores that, in the absence of work, all heat flow results in a change in internal energy. Consequently, any thermal interaction affects the system's energy state without causing mechanical work.