Chapter Review
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Chapter 17: Phases and Phase Changes Chapter Review |
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Chapter Review
This is the second of three chapters on thermodynamics. In this chapter we use the kinetic theory of gases to see how the microscopic properties of a system give rise to the macroscopic properties with which we are more familiar. We also explore the phenomenology of phase changes. The energy needed to bring about a phase change and application of energy conservation to phase changes, and other heat transfer processes, within a system are also treated.
17-1 Ideal Gases
An ideal gas is a gas in which the gas particles (atoms or molecules) do not interact, that is, they move freely and independently of each other. The behavior of ideal gases is a close approximation to the behavior of most real gases. That state of the system of particles making up the gas is determined by the pressure P, temperature T, number of particles N, and volume V of the gas. An equation that shows how these quantities depend on each other is called an equation of state. The detailed study of gases has shown that the equation of state for an ideal gas is
PV = NkT,
where k is a fundamental constant called the Boltzmann constant which has the value
k = 1.38 x 10-23 J/K.
An alternative way to write the equation of state for an ideal gas uses the concept of the mole. A mole is the amount of a substance that contains 6.022 x 1023 entities; this value is called Avogadro's number NA. The masses of atoms (and molecules) are often quoted by stating the mass of one mole of atoms (or molecules) called the atomic mass (or molecular mass) of the substance. The use of moles in the equation of state can be seen by noting that the number of particles in a gas can now be written as N - nNA, where n is the number of moles in the gas. The product of the two constants NAk is another constant called the gas constant R, which has the value
.
Putting these factors together in the equation of state gives
PV = nRT
as a commonly used alternative version of the equation of state for an ideal gas; this latter version is often called the ideal gas law.
The ideal-gas equation of state contains all the information about how the relevant quantities relate. This includes Boyle's law which states that for fixed N and T the product of pressure and volume is constant:
PiVi = PfVf.
Physlet Illustration: Boyle's Law | |
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| A piston sits atop a cylinder filled with an ideal gas. The cylinder is surrounded by a constant temperature bath. Each grid in the animation represents a volume of 10 cm3. A digital temperature probe, which reads the gas temperature in °C, is attached to the cylinder. A digital pressure sensor, which reads the pressure in kPa, is also attached. Play the animation to push the piston down into the cylinder, and see how the volume and pressure change. The graph shows pressure (in kPa) vs. volume (in cm3). Can you verify Boyle's Law? How many moles of gas are in the cylinder? Start | |
Hints:
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Curves that are plotted under the condition of fixed temperature are called isotherms. Similarly, the ideal gas law also incorporates Charles' Law which states that for fixed N and P the ratio V/T is constant:
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Notice that fixed N also implies fixed n.
Physlet Illustration: Charles' Law | |
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| A piston floats atop a cylinder filled with an ideal gas. The cylinder is surrounded by a constant temperature bath. Each grid in the animation represents a volume of 10 cm3. A digital temperature probe, which reads the gas temperature in °C, is attached to the cylinder. A digital pressure sensor, which reads the pressure in kPa, is also attached. Play the animation to heat the temperature bath, and see how the volume changes. The graph shows volume (in cm3) vs. temperature (in Kelvins). Can you verify Charles' Law? How many moles of gas are in the cylinder? Start | |
Hints:
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Exercise 17.1 Inflating a Tire: An automotive worker needs to pump up an empty tire that has an inner volume of 0.0192 m3. If the temperature in the manufacturing plant is 28.5 oC, what will be the gauge pressure in the tire, in psi, if 2.70 moles of air is pumped into it?
Solution: We are given the following information:
Given: V = 0.0192 m3, T = 28.5 oC, n = 2.70 mol Find: P in psi
We can solve this problem by using the ideal-gas equation of state. Since we are given the number of moles of air we will use the form that contains n. Solving this equation for pressure yields
.
Since this expression assumes that T is in Kelvin we must remember to apply the conversion. Doing this conversion gives a pressure of
.
The above result gives the absolute pressure in the tire. What we really want is the gauge pressure which is obtained by subtracting off atmospheric pressure. Thus,
which gives us the final result in pounds per square inch.
Example 17.2 Compressed Air: If you must transfer air from a 3.75 m3 container at atmospheric pressure to a 1.35 m3 container at the same temperature, to what pressure must you compress the air?
Picture the Problem The picture shows the larger container connected to the smaller container by a tube through which the air will pass. [The mechanism that compresses the air is not shown.]
Strategy Both the amount of gas N and the temperature T remain fixed in this problem, so we can use Boyle's law to determine the final pressure.
Solution
| 1. Use Boyle's law to solve for the final pressure: |  |
| 2. Obtain the numerical result: |  |
Insight The fact that the pressure would increase if you squeeze the same gas into a smaller container should also appeal to your intuition. If it doesn't, think about it.
Physlet Illustration: Ideal Gas Law at Constant Volume | |
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| A piston is clamped in place at the top of a cylinder filled with an ideal gas, fixing the volume at 960 cm3. The cylinder is surrounded by a constant temperature bath. A digital temperature probe, which reads the gas temperature in °C, is attached to the cylinder. A digital pressure sensor, which reads the pressure in kPa, is also attached. Play the animation to heat the temperature bath, and see how the pressure changes. The graph shows pressure (in kPa) vs. temperature (in Kelvins). Can you verify the Ideal Gas Law? How many moles of gas are in the cylinder? Start | |
Hints:
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Practice Quiz
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