15.3

The First Law of Thermodynamics

CONCEPTS AT A GLANCE In Chapter 6 we saw that forces can do work and that work can change the kinetic and potential energy of an object. For example, the atoms and molecules of a substance exert forces on one another. As a result, they have kinetic and potential energy. These and other kinds of molecular energy constitute the internal energy of a substance. When a substance participates in a process involving energy in the form of work and heat, the internal energy of the substance can change. The relationship between work, heat, and changes in the internal energy is known as the first law of thermodynamics and is illustrated in the Concepts-at-a-Glance chart in Figure 15-3. We will now see that the first law of thermodynamics is an expression of the conservation of energy.

Figure 15-3 CONCEPTS AT A GLANCE GLANCE The concepts of work, heat, and the internal energy of a system are related by the first law of thermodynamics, which is an expression of the law of conservation of energy. The body of this basketball player contains internal energy. Part of this energy is used to do the work of playing the game, and part is lost in the form of heat carried off by evaporating sweat. (Tim Pannell/Masterfile) Problem solving insight

Problem solving insight

Suppose that a system gains heat Q and that this is the only effect occurring. Consistent with the law of conservation of energy, the internal energy of the system increases from an initial value of Ui to a final value of Uf, the change being . In writing this equation, we use the convention that heat Q is positive when the system gains heat and negative when the system loses heat. The internal energy of a system can also change because of work. If a system does work W on its surroundings and there is no heat flow, energy conservation indicates that the internal energy of the system decreases from Ui to Uf, the change now being . The minus sign is included because we follow the convention that work is positive when it is done by the system and negative when it is done on the system. A system can gain or lose energy simultaneously in the form of heat Q and work W. The change in internal energy due to both factors is given by Equation 15.1. Thus, the first law of thermodynamics is just the conservation-of-energy principle applied to heat, work, and the change in the internal energy.

| The First Law of Thermodynamics
| The internal energy of a system changes from an initial value Ui to a final value of Uf due to heat Q and work W:

| (15.1)

Q is positive when the system gains heat and negative when it loses heat. W is positive when work is done by the system and negative when work is done on the system.

Example 1 illustrates the use of Equation 15.1 and the sign conventions for Q and W.

| Example 1 | | | Positive and Negative Work

Figure 15-4 illustrates a system and its surroundings. In part a, the system gains 1500 J of heat from its surroundings, and 2200 J of work is done by the system on the surroundings. In part b, the system also gains 1500 J of heat, but 2200 J of work is done on the system by the surroundings. In each case, determine the change in the internal energy of the system.

Figure 15-4 (a) The system gains energy in the form of heat but loses energy because work is done by the system. (b) The system gains energy in the form of heat and also gains energy because work is done on the system.

Reasoning In Figure 15-4a the system loses more energy in doing work than it gains in the form of heat, so the internal energy of the system decreases. Thus, we expect the change in the internal energy, , to be negative. In part b of the drawing, the system gains energy in the form of both heat and work. The internal energy of the system increases, and we expect to be positive.
Solution (a) The heat is positive, , since it is gained by the system. The work is positive, , since it is done by the system. According to the first law of thermodynamics

| (15.1)

The minus sign for indicates that the internal energy has decreased, as expected.
(b) The heat is positive, , since it is gained by the system. But the work is negative, , since it is done on the system. Thus,

| (15.1)

The plus sign for indicates that the internal energy has increased, as expected.

Problem solving insight When using the first law of thermodynamics, as expressed by Equation 15.1, be careful to follow the proper sign conventions for the heat Q and the work W.

In the first law of thermodynamics, the internal energy U, heat Q, and work W are energy quantities, and each is expressed in energy units such as joules. However, there is a fundamental difference between U, on the one hand, and Q and W on the other. The next example sets the stage for explaining this difference.

| Example 2 | | | An Ideal Gas

The temperature of three moles of a monatomic ideal gas is reduced from to by two different methods. In the first method 5500 J of heat flows into the gas, whereas in the second, 1500 J of heat flows into it. In each case find (a) the change in the internal energy and (b) the work done by the gas.
Reasoning Since the internal energy of a monatomic ideal gas is (Equation 14.7) and since the number of moles n is fixed, only a change in temperature T can alter the internal energy. Because the change in T is the same in both methods, the change in U is also the same. From the given temperatures, the change in internal energy can be determined. Then, the first law of thermodynamics can be used with and the given heat values to calculate the work for each of the methods.
Solution (a) Using Equation 14.7 for the internal energy of a monatomic ideal gas, we find for each method of adding heat that

(b) Since is now known and the heat is given in each method, Equation 15.1 can be used to determine the work:

In each method the gas does work, but it does more in the first method.

Problem solving insight

To understand the difference between U and either Q or W, consider the value for in Example 2. In both methods is the same. Its value is determined once the initial and final temperatures are specified because the internal energy of an ideal gas depends only on the temperature. Temperature is one of the variables (along with pressure and volume) that define the state of a system. The internal energy depends only on the state of a system, not on the method by which the system arrives at a given state. In recognition of this characteristic, internal energy is referred to as a function of state.* In contrast, heat and work are not functions of state because they have different values for each different method used to make the system change from one state to another, as in Example 2.

Check Your Understanding 1

A gas is enclosed within a chamber that is fitted with a frictionless piston. The piston is then pushed in, thereby compressing the gas. Which statement below regarding this process is consistent with the first law of thermodynamics?(The answer is given at the end of the book.)

a. | The internal energy of the gas will increase.
b. | The internal energy of the gas will decrease.
c. | The internal energy of the gas will not change.
d. | The internal energy of the gas may increase, decrease, or remain the same, depending on the amount of heat that the gas gains or loses.

Background: Answering this question depends on an understanding of the first law of thermodynamics. This law relates the concepts of internal energy, heat, and work.
For similar questions (including calculational counterparts), consult Self-Assessment Test 15.1. This test is described at the end of Section 15.6.

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