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Chapter 15: Problem 4

A system does \(4.8 \times 10^{4} \mathrm{J}\) of work, and \(7.6 \times 10^{4} \mathrm{J}\) of heat flows into the system during the process. Find the change in the internal energy of the system.

Short Answer

Expert verified
The change in internal energy is \( 2.8 \times 10^{4} \mathrm{J} \).

Step by step solution

01

Understanding the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in the internal energy of a system, \( \Delta U \), is equal to the heat added to the system, \( Q \), minus the work done by the system, \( W \). This relationship can be expressed with the formula: \( \Delta U = Q - W \). Understanding this formula is essential for solving the problem.
02

Identify and Substitute Known Values

From the exercise, we know that the heat added to the system, \( Q \), is \( 7.6 \times 10^{4} \mathrm{J} \) and the work done by the system, \( W \), is \( 4.8 \times 10^{4} \mathrm{J} \). We will substitute these values into the formula \( \Delta U = Q - W \).
03

Calculate Change in Internal Energy

Now, substitute the values into the formula: \( \Delta U = 7.6 \times 10^{4} \mathrm{J} - 4.8 \times 10^{4} \mathrm{J} \). Perform the subtraction to find \( \Delta U \).
04

Perform the Subtraction

Calculate \( 7.6 \times 10^{4} - 4.8 \times 10^{4} \), which gives us \( 2.8 \times 10^{4} \mathrm{J} \). This is the change in internal energy of the system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Internal Energy Change
The concept of internal energy change is crucial in understanding how energy moves within a system. The internal energy of a system is the total energy contained within it, which includes kinetic and potential energy of the particles composing the system.
According to the First Law of Thermodynamics, the change in internal energy, denoted as \( \Delta U \), can be calculated when you know the amount of heat transferred and the work done.
The formula to calculate this change is \( \Delta U = Q - W \), where:
  • \( Q \) is the heat added to or removed from the system.
  • \( W \) is the work done by or on the system.
In the provided problem, understanding how changes in heat and work affect \( \Delta U \) helps illustrate energy conservation within a thermodynamic process.
Heat and Work
Heat and work are two fundamental ways energy is transferred between systems and their surroundings.
  • Heat (\( Q \)): Heat energy flows due to temperature difference. It can be thought of as energy in transit, moving from a hotter to a cooler body. In this exercise, \( 7.6 \times 10^{4} \, \text{J} \) of heat flows into the system, increasing its internal energy.

  • Work (\( W \)): Work is done when an external force causes an object to move. In thermodynamic systems, work involves volume changes. The system does \( 4.8 \times 10^{4} \, \text{J} \) of work, implying it uses some of its energy to perform tasks outside.
Understanding the interaction of heat and work is essential in applying the First Law of Thermodynamics, as these quantities contribute to the internal energy change.
Thermodynamic System Analysis
Analyzing a thermodynamic system involves understanding the interactions within a defined boundary, referred to as the system, and its surroundings. This analysis helps in applying the First Law of Thermodynamics effectively.

A standard approach is to track energy inputs and outputs:
  • System Boundary: Defines what's inside to focus on (e.g., gas in a piston).
  • Surroundings: Everything outside the boundary influencing or influenced by the system.
  • State Functions: Internal energy, pressure, and volume are examples that describe the system鈥檚 state.
In this exercise, establishing a clear view of how energy flows into the system as heat and out as work helps in calculating the internal energy change. Such assessments show the state transformations a system undergoes and utilizes the First Law to ensure energy conservation principles are maintained.

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Most popular questions from this chapter

A Carnot heat pump operates between an outdoor temperature of \(265 \mathrm{K}\) and an indoor temperature of \(298 \mathrm{K}\). Find its coefficient of performance.

A power plant taps steam superheated by geothermal energy to \(505 \mathrm{K}\) (the temperature of the hot reservoir) and uses the steam to do work in turning the turbine of an electric generator. The steam is then converted back into water in a condenser at \(323 \mathrm{K}\) (the temperature of the cold reservoir), after which the water is pumped back down into the earth where it is heated again. The output power (work per unit time) of the plant is 84000 kilowatts. Determine (a) the maximum efficiency at which this plant can operate and (b) the minimum amount of rejected heat that must be removed from the condenser every twenty-four hours.

Due to a tune-up, the efficiency of an automobile engine increases by \(5.0 \% .\) For an input heat of \(1300 \mathrm{J},\) how much more work does the engine produce after the tune-up than before?

A system undergoes a two-step process. In the first step, the internal energy of the system increases by \(228 \mathrm{J}\) when \(166 \mathrm{J}\) of work is done on the system. In the second step, the internal energy of the system increases by \(115 \mathrm{J}\) when 177 J of work is done on the system. For the overall process, find the heat. What type of process is the overall process? Explain.

The sublimation of zinc (mass per mole \(=0.0654 \mathrm{kg} / \mathrm{mol}\) ) takes place at a temperature of \(6.00 \times 10^{2} \mathrm{K},\) and the latent heat of sublimation is \(1.99 \times 10^{6} \mathrm{J} / \mathrm{kg} .\) The pressure remains constant during the sublimation. Assume that the zinc vapor can be treated as a monatomic ideal gas and that the volume of solid zinc is negligible compared to the corresponding vapor. Concepts: (i) What is sublimation, and what is the latent heat of sublimation? (ii) When a solid phase changes to a gas phase, does the volume of the material increase or decrease, and by how much? (iii) As the material changes from a solid to a gas, does it do work on the environment, or does the environment do work on it? How much work is involved? (iv) In this problem we begin with heat \(Q\) and realize that it is used for two purposes: First, it makes the solid change into a gas, which entails a change \(\Delta U\) in the internal energy of the material, \(\Delta U=U_{\mathrm{gas}}-U_{\text {solid }} .\) Second, it allows the expanding material to do work \(W\) on the environment. According to the conservation-of- energy principle, how is \(Q\) related to \(\Delta U\) and \(W ?\) (v) According to the first law of thermodynamics, how is \(Q\) related to \(\Delta U\) and \(W\) ? Calculations: What is the change in the internal energy of zinc when \(1.50 \mathrm{kg}\) of zinc sublimates?
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