Problem 23 - Pearson Physics

Question

Scrutinize State whether each of the following statements is true or false. If a statement is false, revise it so that it becomes true. (a) The change in thermal energy in a constant-volume process is zero. (b) The work done in a constant-pressure process is zero. (c) The area under the curve on a pressure-volume graph is equal to the work. (d) Work and heat are equal in an isothermal process. (e) The thermal energy of a system increases when work is done on it in an adiabatic process.

Step-by-Step Solution

Answer

(a) False - Change depends on heat; (b) False - Work can occur with volume change; (c) True; (d) False - Heat and work equal in magnitude, with sign; (e) True - Thermal energy increases with work done on the system.

Step 1: Evaluate Statement (a)

For a constant-volume process, the volume does not change, i.e., \( \Delta V = 0 \). The change in thermal energy \( \Delta E_{th} \) relates to heat (\( Q \)) and work (\( W \)). Since there is no volume change, no work is done (\( W = P \Delta V = 0 \)). Hence, the change in thermal energy, dependent only on heat, is not necessarily zero. Revised statement: The change in thermal energy in a constant-volume process is not necessarily zero but is solely due to heat exchange.

Step 2: Evaluate Statement (b)

In a constant-pressure process, the work done is given by the formula \( W = P \Delta V \). If the process is not isochoric (i.e., \( \Delta V eq 0 \)), work is performed. Therefore, the statement is false.Revised statement: The work done in a constant-pressure process is not zero if there is a change in volume.

Step 3: Evaluate Statement (c)

On a pressure-volume graph, the area under the curve represents the work done during the process. This is a true statement, as work in thermodynamics is calculated by the integral of pressure with respect to volume.

Step 4: Evaluate Statement (d)

In an isothermal process for an ideal gas, the change in internal energy is zero (\( \Delta U = 0 \)), so the heat added to the system (\( Q \)) equals the work done by the system (\( W \)). This usually implies \( Q = -W \), not equal in value. Hence, the initial statement is incorrect for clarity.Revised statement: In an isothermal process, the heat added to the system equals the work done by the system in magnitude if we account for the sign.

Step 5: Evaluate Statement (e)

In an adiabatic process, no heat is exchanged (\( Q = 0 \)). Therefore, if work is done on the system, the thermal energy increases according to the first law of thermodynamics \(( \Delta U = Q - W )\), implying that \( \Delta U = - W \), making the statement true since work done on the system increases thermal energy.

Key Concepts

Constant-volume process

In a constant-volume process, the volume of the system remains unchanged. This means that \( \Delta V = 0 \). As a result, no mechanical work is done on or by the system because work in thermodynamics is calculated using the formula \( W = P \Delta V \). With \( \Delta V = 0 \), \( W = 0 \).

However, the thermal energy of the system, denoted as \( \Delta E_{th} \), can still change. In a constant-volume process, the change in thermal energy is conducted solely through heat exchange \( Q \). Therefore:

• The thermal energy change is not necessarily zero.
• It is exclusively reliant on heat transfer, not work.

Understanding this helps differentiate how heat and work affect system energy.

Isothermal process

An isothermal process occurs at a constant temperature. For an ideal gas, the internal energy is solely a function of temperature. This means that in an isothermal process, where temperature is constant, there is no change in internal energy \( \Delta U = 0 \).

In such a case, any heat added to the system \( Q \) is perfectly converted into work done by the system \( W \), resulting in the equation \( Q = W \). Though one might think that work and heat are equal in value and sign, the statement needs a careful understanding of signs, where \( Q = -W \) balances the equation

• This process is isothermal due to constant temperature.
• The internal energy remains unchanged.
• Heat input translates directly into work output, balancing the equation.

Grasping this helps illustrate the unique nature of isothermal transformations.

Adiabatic process

During an adiabatic process, the system is thermally insulated, meaning no heat enters or leaves the system (\( Q = 0 \)). Thus, changes in a system's internal energy come solely from work done on or by the system.

When work is done on the system, it results in an increase in the system's thermal energy because of the relationship expressed by the first law of thermodynamics: \[ \Delta U = Q - W = -W \].

• No heat exchange with surroundings, making it adiabatic.
• Thermal energy increases if work is done on the system.
• The equation shows that work done affects internal energy directly.

Understanding adiabatic processes is essential for thorough thermodynamics comprehension.

Pressure-volume graph

A pressure-volume graph, often used in thermodynamics, displays a system undergoing changes in pressure and volume. The area under the curve in this graph represents the work done during the process. This follows from the work formula in thermodynamics: \( W = \int{P\, dV} \).
When you look at a pressure-volume graph:

• The curve shape indicates if the process is isothermal, adiabatic, or constant-volume.
• The larger the area, the more work is done.
• Different processes have unique curve characteristics.

Interpreting these graphs is a skill that allows for the visualization and quantification of work.

Work and heat interaction

In thermodynamics, work and heat are two ways energy is transferred into or out of a system. Work is the mechanical energy transfer due to force (\( W = \int{P\, dV} \)), while heat is energy transfer due to temperature differences.
Understanding their interaction involves recognizing:

• Heat flows from high to low temperature regions.
• Work is done when there is a force causing displacement.
• They are not always equal but can be related through conservation laws.

Recognizing how they relate and differ is key to understanding energy exchanges in different types of processes.