Problem 53 - University Physics with Modern Physics

Question

In a certain process, \(2.15 \times 10^{5} \mathrm{J}\) of heat is liberated by a system, and at the same time the system contracts under a constant external pressure of \(9.50 \times 10^{5} \mathrm{Pa}\) . The internal energy of the system is the same at the beginning and end of the process. Find the change in volume of the system. (The system is not an ideal gas.)

Step-by-Step Solution

Answer

The change in volume is -0.2263 m³.

Step 1: Understanding the First Law of Thermodynamics

The problem involves heat, work, and change in volume. Use the first law of thermodynamics: \[ \Delta U = Q - W \]where \( \Delta U \) is the change in internal energy, \( Q \) is the heat exchanged, and \( W \) is the work done by the system. Since the internal energy does not change in this process (\( \Delta U = 0 \)), we have:\[ 0 = Q - W \].

Step 2: Relate Work to Pressure and Change in Volume

Work done by the system can also be expressed as the product of pressure and change in volume:\[ W = P \Delta V \]Substitute this into the modified first law equation:\[ Q = P \Delta V \]

Step 3: Solve for Change in Volume

Rearrange the equation to solve for \( \Delta V \):\[ \Delta V = \frac{Q}{P} \]Substitute the given values:\[ Q = -2.15 \times 10^{5} \text{ J} \] (heat is liberated, hence negative) and \( P = 9.50 \times 10^{5} \text{ Pa} \).

Step 4: Calculate the Change in Volume

Use the formula:\[ \Delta V = \frac{-2.15 \times 10^{5}}{9.50 \times 10^{5}} = -0.2263 \] Thus, the change in volume \( \Delta V \) is \(-0.2263 \text{ m}^3\). The negative sign indicates a decrease in volume.

Key Concepts

Change in Volume

In thermodynamics, the change in volume is an important concept, especially when analyzing systems under pressure. When a system undergoes a process, its volume can expand or contract depending on the pressure and the heat exchange involved. In this particular scenario, we are tasked with finding the change in volume of a system that contracts under constant external pressure.

Utilizing the First Law of Thermodynamics, we can connect the change in volume to the work done by the system. According to the formula, work is the product of pressure and change in volume: \[ W = P \Delta V \] In the exercise, we've established that the internal energy does not change, leading us to the equation: \[ Q = P \Delta V \] Here, work done is considered in terms of pressure and change in volume. By rearranging this equation, the change in volume can be expressed as: \[ \Delta V = \frac{Q}{P} \] This formula allows us to calculate how much a system's volume changes during a process.

The negative sign in the resulting \( \Delta V \) value indicates a reduction in volume, showing that the system contracts rather than expands. Understanding this concept is essential, especially in classical thermodynamics where such calculations frequently occur when analyzing gas laws, engine efficiency, and thermodynamic cycles.

Internal Energy

Internal energy is a fundamental concept in thermodynamics representing the total energy contained within a system. It encompasses all forms of kinetic and potential energy associated with the random motion and interactions of molecules. When analyzing any thermodynamic process, internal energy serves as a key variable.

The First Law of Thermodynamics, described by the equation: \[ \Delta U = Q - W \] relates changes in internal energy \( \Delta U \) to heat exchange \( Q \) and work \( W \). In this particular exercise, the internal energy remains constant throughout the process, meaning \( \Delta U = 0 \). This simplifies our calculations and tells us that any heat liberated or absorbed in the process must be equal and opposite to the work done by the system.

Since the system does not exhibit any change in its internal energy, it signifies that the entire amount of heat removed is converted into mechanical work against an external pressure. This is an illustration of energy conservation in isolated systems, and it highlights the power of the First Law of Thermodynamics in predicting system behavior during various physical processes.

Heat Exchange

Heat exchange refers to the transfer of thermal energy between a system and its surroundings. In thermodynamics, heat is a form of energy transfer that occurs due to a temperature difference between a system and its environment. It can be positive when absorbed by the system or negative when released.

In the stated problem, the system liberates heat, which means that heat is being transferred from the system to its surroundings. In thermodynamic terms, this is expressed as a negative heat value, \( Q = -2.15 \times 10^5 \text{ J} \). The negative sign signifies an exothermic process, where energy is emitted by the system.

Understanding heat exchange is crucial because it directly affects the internal energy and work performed by a system. As discussed in the First Law of Thermodynamics, heat exchange is one of the components that governs the energy balance within a system. In this case, since no change in internal energy is noted, the liberated heat is fully utilized for performing work on the system, specifically in changing its volume. Recognizing how heat and work interact allows us to comprehensively analyze energy exchanges in a wide range of physical and chemical processes.