# 2: Gas Laws

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Early experimenters discovered that the pressure, volume, and temperature of a gas are related by simple equations. The classical gas laws include Boyle’s law, Charles’ law, Avogadro’s hypothesis, Dalton’s law of partial pressures, and Amagat’s law of partial volumes. These laws were inferred from experiments done at relatively low pressures and at temperatures well above those at which the gases could be liquefied. We begin our discussion of gas laws by reviewing the experimental results that are obtained under such conditions. As we extend our experiments to conditions in which gas densities are greater, we find that the accuracy of the classical gas laws decreases.

• 2.1: Boyle's Law
Robert Boyle discovered Boyle’s law in 1662. Boyle’s discovery was that the pressure, P, and volume, V, of a gas are inversely proportional to one another if the temperature, T, is held constant. We can imagine rediscovering Boyle’s law by trapping a sample of gas in a tube and then measuring its volume as we change the pressure.
• 2.2: Charles' Law
Charles’ law relates the volume and temperature of a gas when measurements are made at constant pressure. We can imagine rediscovering Charles’ law by trapping a sample of gas in a tube and measuring its volume as we change the temperature, while keeping the pressure constant. This presumes that we have a way to measure temperature, perhaps by defining it in terms of the volume of a fixed quantity of some other fluid—like liquid mercury.
Avogadro’s hypothesis is another classical gas law. It can be stated: At the same temperature and pressure, equal volumes of different gases contain the same number of molecules. When the mass, in grams, of an ideal gas sample is equal to the gram molar mass (traditionally called the molecular weight) of the gas, the number of molecules in the sample is equal to Avogadro’s number.
There are numerous ways to measure Avogadro’s number. One such method is to divide the charge of one mole of electrons by the charge of a single electron. We can obtain the charge of a mole of electrons from electrolysis experiments. The charge of one electron can be determined in a famous experiment devised by Robert Millikan, the “Millikan oil-drop experiment”.
• 2.5: The Kelvin Temperature Scale
• 2.6: Deriving the Ideal Gas Law from Boyle's and Charles' Laws
• 2.7: The Ideal Gas Constant and Boltzmann's Constant
• 2.8: Real Gases Versus Ideal Gases
We imagine that the results of a large number of experiments are available for our analysis. Our characterization of these results has been that all gases obey the same equations—Boyle’s law, Charles’ law, and the ideal gas equation—and do so exactly. This is an oversimplification. In fact they are always approximations. They are approximately true for all gases under all “reasonable” conditions, but they are not exactly true for any real gas under any condition.
• 2.9: Temperature and the Ideal Gas Thermometer
We can define temperature in terms of the expansion of any constant-pressure gas that behaves ideally. In principle, we can measure the same temperature using any gas, so long as the constant operating pressure is low enough. When we do so, our device is called the ideal gas thermometer. In so far as any gas behaves as an ideal gas at a sufficiently low pressure, any real gas can be used in an ideal gas thermometer and to measure any temperature accurately.
• 2.10: Deriving Boyle's Law from Newtonian Mechanics
We can derive Boyle’s law from Newtonian mechanics. This derivation assumes that gas molecules behave like point masses that do not interact with one another. The pressure of the gas results from collisions of the gas molecules with the walls of the container. The contribution of one collision to the force on the wall is equal to the change in the molecule’s momentum divided by the time between collisions. The magnitude of this force depends on the molecule’s speed and angle it strikes the wall.
• 2.11: The Barometric Formula
We can measure the pressure of the atmosphere at any location by using a barometer. A mercury barometer is a sealed tube that contains a vertical column of liquid mercury. The space in the tube above the liquid mercury is occupied by mercury vapor. Since the vapor pressure of liquid mercury at ordinary temperatures is very low, the pressure at the top of the mercury column is very low and can usually be ignored.
• 2.12: Van der Waals' Equation
We often assume that gas molecules do not interact with one another, but simple arguments show that this can be only approximately true. Real gas molecules must interact with one another. At short distances they repel one another. At somewhat longer distances, they attract one another. Van der Waals’ equation fits pressure-volume-temperature data for a real gas better than the ideal gas equation does. The improved fit is obtained by introducing two experimentally determined parameters.
• 2.13: Virial Equations
Expanding the compressibility factor to a polynomial in the pressure results in a better description of real gas behavior. The values for the parameters of this expansion are often tabulated for each gas independently.
• 2.14: Gas Mixtures - Dalton's Law of Partial Pressures
Thus far, our discussion of the properties of a gas has implicitly assumed that the gas is pure. We turn our attention now to mixtures of gases—gas samples that contain molecules of more than one compound. Mixtures of gases are common, and it is important to understand their behavior in terms of the properties of the individual gases that make it up. The ideal-gas laws we have for mixtures are approximations. Fortunately, these approximations are often very good.
• 2.15: Gas Mixtures - Amagat's Law of Partial Volums
Amagat’s law of partial volumes asserts that the volume of a mixture is equal to the sum of the partial volumes of its components.
• 2.16: Problems

$${}^{1}$$$${}^{\ }$$We use the over-bar to indicate that the quantity is per mole of substance. Thus, we write $$\overline{N}$$ to indicate the number of particles per mole. We write $$\overline{M}$$ to represent the gram molar mass. In Chapter 14, we introduce the use of the over-bar to denote a partial molar quantity; this is consistent with the usage introduced here, but carries the further qualification that temperature and pressure are constant at specified values. We also use the over-bar to indicate the arithmetic average; such instances will be clear from the context.

$${}^{2}$$$${}^{\ }$$The unit of temperature is named the kelvin, which is abbreviated as K.

$${}^{3}$$$${}^{\ }$$A redefinition of the size of the unit of temperature, the kelvin, is under consideration. The practical effect will be inconsequential for any but the most exacting of measurements.

$${}^{4}$$$${}^{\ }$$For a thorough discussion of the development of the concept of temperature, the evolution of our means to measure it, and the philosophical considerations involved, see Hasok Chang, Inventing Temperature, Oxford University Press, 2004.

$${}^{5}$$$${}^{\ }$$See T. L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley Publishing Company, 1960, p 286.

$${}^{6}$$$${}^{\ }$$See S. M. Blinder, Advanced Physical Chemistry, The Macmillan Company, Collier-Macmillan Canada, Ltd., Toronto, 1969, pp 185-18926

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