14.5

Concepts & Calculations

This chapter introduces the ideal gas law, which is a relation between the pressure, volume, temperature, and number of moles of an ideal gas. In Section 10.1, we examined how the compression of a spring depends on the force applied to it. Concepts & Calculations Example 9 reviews how a gas produces a force and why an ideal gas at different temperatures causes a spring to compress by different amounts.

| Concepts & Calculations Example 9 | | | The Ideal Gas Law and Springs

Figure 14-18 shows three identical chambers containing a piston and a spring whose spring constant is . The chamber in part a is completely evacuated, and the piston just touches its left end. In this position, the spring is unstrained. In part b of the drawing, 0.75 mol of ideal gas 1 is introduced into the chamber, and the spring compresses by . In part c, 0.75 mol of ideal gas 2 is introduced into the chamber, and the spring compresses by . Find the temperature of each gas.

Figure 14-18 Two ideal gases at different temperatures compress the spring by different amounts.

Concept Questions and Answers Which gas exerts the greater force on the piston?

Answer We know that a greater force is required to compress a spring by a greater amount. Therefore, gas 2 exerts the greater force.

How is the force required to compress a spring related to the displacement of the spring from its unstrained position?

Answer According to Equation 10.1, the applied force FxApplied required to compress a spring is directly proportional to the displacement x of the spring from its unstrained position; , where k is the spring constant of the spring.

Which gas exerts the greater pressure on the piston?

Answer According to Equation 11.3, pressure is defined as the magnitude F of the force acting perpendicular to the surface of the piston divided by the area A of the piston; . Since gas 2 exerts the greater force, and the area of the piston is the same for both gases, gas 2 exerts the greater pressure.

Which gas has the greater temperature?

Answer According to the ideal gas law, , gas 2 has the greater temperature. Both gases contain the same number n of moles. However, gas 2 has both a greater pressure P and volume V. Thus, it has the greater temperature.

Solution We can use the ideal gas law in the form to determine the temperature T of each gas. First, however, we must find the pressure. According to Equation 11.3, the pressure is the magnitude F of the force that the gas exerts on the piston divided by the area A of the piston, so . Recall that the force FxApplied applied to the piston to compress the spring is related to the displacement x of the spring by (Equation 10.1). Thus, , and the pressure can be written as . Using this expression for the pressure in the ideal gas law gives

However, the cylindrical volume V of the gas is equal to the product of the distance x and the area A, so . With this substitution, the temperature of the gas becomes

The temperatures of the gases are

As anticipated, gas 2, which compresses the spring more, has the higher temperature.

The kinetic theory of gases is important because it allows us to understand the relation between the macroscopic properties of a gas, such as pressure and temperature, and the microscopic properties of its particles, such as speed and mass. The following example reviews the essential features of this theory.

| Concepts & Calculations Example 10 | | | Hydrogen Atoms in Outer Space

In outer space the density of matter is extremely low, about one atom per cm3. The matter is mainly hydrogen atoms whose rms speed is 260 m/s. A cubical box, 2.0 m on a side, is placed in outer space, and the hydrogen atoms are allowed to enter. (a) What is the magnitude of the force that the atoms exert on one wall of the box? (b) Determine the pressure that the atoms exert. (c) Does outer space have a temperature, and, if so, what is it?
Concept Questions and Answers Why do hydrogen atoms exert a force on the walls of the box?

Answer Every time an atom collides with a wall and rebounds, the atom exerts a force on the wall. Imagine opening your hand so it is flat and having someone throw a ball straight at it. Your hand is like the wall, and as the ball rebounds, you can feel the force. Intuitively, you would expect the force to become greater as the speed and mass of the ball become greater. This is indeed the case.

Do the atoms generate a pressure on the walls of the box?

Answer Yes. Pressure is defined as the magnitude of the force exerted perpendicularly on a wall divided by the area of the wall. Since the atoms exert a force, they also produce a pressure.

Do hydrogen atoms in outer space have a temperature? If so, how is the temperature related to the microscopic properties of the atoms?

Answer Yes. The Kelvin temperature is proportional to the average kinetic energy of an atom. The average kinetic energy, in turn, is proportional to the mass of an atom and the square of the rms speed. Since we know both these quantities, we can determine the temperature of the gas.

Solution

a. | The magnitude F of the force exerted on a wall is given by (see Section 14.3)

where N is the number of atoms in the box, m is the mass of a single atom, vrms is the rms speed of the atoms, and L is the length of one side of the box. The volume of the cubical box is . The number N of atoms is equal to the number of atoms per cubic centimeter times the volume of the box in cubic centimeters:

The magnitude of the force acting on one wall is

b. | The pressure is the magnitude of the force divided by the area A of a wall:

c. | According to Equation 14.6, the Kelvin temperature T of the hydrogen atoms is related to the average kinetic energy of an atom by where k is Boltzmann’s constant. Solving this equation for the temperature gives

This is a frigid 2.7 kelvins above absolute zero.

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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