chapter 10: gases - sogang

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Chapter 10Gases

▶ THE TRANSIT OF VENUS occurs when Venus passes directly between the Earth and the Sun. The small black circle in the upper right is the shadow of Venus passing in front of the Sun.

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What’s Ahead10.1 CHARACTERISTICS OF GASES

We begin by comparing the distinguishing characteristics of gases with those of liquids and solids.

10.2 PRESSUREWe then study gas pressure, how it is measured, and the units used to express it, as well as consider Earth’s atmosphere and the pressure it exerts.

10.3 THE GAS LAWSWe see that the state of a gas can be expressed in terms of its volume, pressure, temperature, and quantity and examine several gas laws, which are empirical relationships among these four variables.

10.4 THE IDEAL-GAS EQUATIONWe find that the gas laws yield the ideal-gas equation, PV = nRT. Although this equation is not obeyed exactly by any real gas, most gases come very close to obeying it at ordinary temperatures and pressures.

10.5 FURTHER APPLICATIONS OF THE IDEAL-GAS EQUATIONWe use the ideal-gas equation in many calculations, such as the calculation of the density or molar mass of a gas.

Gases

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What’s Ahead10.6 GAS MIXTURES AND PARTIAL PRESSURES

We recognize that in a mixture of gases, each gas exerts a pressure that is part of the total pressure. This partial pressure is the pressure the gas would exert if it were by itself.

10.7 THE KINETIC-MOLECULAR THEORY OF GASESWe see that this theory helps us understand gas behavior on the molecular level. According to the theory, the atoms or molecules that make up a gas move with an average kinetic energy that is proportional to the gas temperature.

10.8 MOLECULAR EFFUSION AND DIFFUSIONWe observe that the kinetic-molecular theory helps us account for such gas properties as effusion, movement through tiny openings, and diffusion, movement through another substance.

10.9 REAL GASES: DEVIATIONS FROM IDEAL BEHAVIORWe learn that real gases deviate from ideal behavior because the gas molecules have finite volume and because attractive forces exist between molecules. The van der Waals equation gives an accurate account of real gas behavior at high pressures and low temperatures.

Gases

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Characteristics of Gases

• Gases behave quite similarly as far as their physical properties are concerned.– Ex) Air (N2 (78%), O2 (21%), and Ar (0.9%)): N2 and O2 behave

physically as one gaseous material because their physical properties are essentially identical.

• Composed mainly of nonmetallic elements with simple formulas and low molar masses.

• Only a few elements exist as gases under ordinary conditions.– H2, N2, O2, F2, Cl2 and noble gases (He, Ne, Ar, Kr, and Xe).– Cf) Vapors: Substances that are liquids or solids under ordinary

conditions, but exist in the gaseous state (e.g. water vapor).

10.1 CHARACTERISTICS OF GASES

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Characteristics of Gases

• Gases differ significantly from solids and liquids.– Gas molecules are relatively far apart; each molecule behaves largely

as though the others are not present.– Gases are readily compressible and expansible.– Gases have extremely low density.

• Gases form homogeneous mixtures with other gases.

10.1 CHARACTERISTICS OF GASES

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Characteristics of Gases10.1 CHARACTERISTICS OF GASES

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Properties Which Define the State of a Gas Sample

1) Temperature2) Pressure3) Volume4) Amount of gas, usually expressed as

number of moles

Having already discussed three of these, we need to define pressure.

10.2 PRESSURE

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Atmospheric Pressure and the Barometer

• Pressure is the amount of force applied to an area:

• Atmospheric pressure is the weight of air per unit of area.

10.2 PRESSURE

Figure 10.1 Calculating atmospheric pressure.

• The SI unit of pressure is the pascal(Pa): 1 Pa = 1 N/m2.

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• A related pressure unit.– Bar: The atmospheric pressure at sea level, 1 bar = 105 Pa = 100 kPa.– Pounds per square inch (psi, lbs/in.2): At sea level, atmospheric

pressure is 14.7 psi.

10.2 PRESSURE

Figure 10.2 A mercury barometer.

• Barometer.– In the 17th century, people believed

that the atmosphere had no weight.– Torricelli invented the barometer and

proved that the atmosphere has weight.– Pascal measured the height of the

mercury column at two different places (the top and the base of a mountain) and his experiment supported Torricelli’s explanation.

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• Standard atmospheric pressure.– The typical pressure at sea level.– The pressure sufficient to support a column of mercury 760 mm high.– In SI units this pressure is 1.01325 × 105 Pa.

10.2 PRESSURE

Figure 10.2 A mercury barometer.

– Standard atmospheric pressure defines some common non–SI units (the atmosphere (atm) and the millimeter of mercury (mm Hg) called the torr (1 torr = 1 mm Hg)).

– mmHg or torr: The difference in the heights measured in mm (h) of two connected columns of mercury.

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• Manometer: A device used to measure the difference in pressure between atmospheric pressure and that of a gas in a vessel.

10.2 PRESSURE

Figure 10.3 A mercury manometer.

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The Pressure–Volume Relationship: Boyle’s Law

• Boyle’s Law: The volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure.

10.3 THE GAS LAWS

Figure 10.4 As a balloon rises in the atmosphere, its volume increases.

Figure 10.5 Boyle’s experiment relating pressure and volume for a gas.

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The Pressure–Volume Relationship: Boyle’s Law

• Boyle’s Law: The volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure.

10.3 THE GAS LAWS

Figure 10.6 Boyle’s Law. For a fixed quantity of gas at constant temperature, the volume of the gas is inversely proportional to its pressure.

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The Temperature–Volume Relationship:Charles’s Law

• Charles’s law: The volume of a fixed amount of gas maintained at constant pressure is directly proportional to its absolute temperature.

10.3 THE GAS LAWS

Figure 10.7 The effect of temperature on volume.

Figure 10.8 Charles’s Law. For a fixed quantity of gas at constant pressure, the volume of the gas is proportional to its temperature.

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The Quantity–Volume Relationship: Avogadro’s Law

• Avogadro’s hypothesis: Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

• Avogadro’s law: The volume of a gas maintained at constant temperature and pressure is directly proportional to the number of moles of the gas.

10.3 THE GAS LAWS

Figure 10.9 Avogadro’s hypothesis. At the same volume, pressure, and temperature, samples of different gases have the same number of molecules but different masses.

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The Ideal-Gas Equation

• The Ideal-Gas Equation.– So far we’ve seen that

– Combining these, we get .

– If we call the proportionality constant R, we obtain an equality:

– Rearranging this, we get the ideal-gas equation (also called the ideal-gas law).

10.4 THE IDEAL-GAS EQUATION

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– Ideal gas: A hypothetical gas whose pressure, volume, and temperature relationships are described completely by the ideal-gas equation.

– The term R in the ideal-gas equation is the gas constant.

– The value for T in the ideal-gas equation must always be the absolute temperature (in kelvins instead of degrees Celsius).


• R = 0.08206 L-atm/mol-K= 8.314 J/mol-K

• The value and units of R depend on the units of P, V, n, and T.

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– Molar volume at standard temperature and pressure (STP): The volume occupied by 1 mol of a gas at STP.• Standard temperature and pressure (STP): The conditions 0 ˚C

(273.15 K) and 1 atm.• The molar volume of an ideal gas at STP is 22.41 L.

• One mole of various real gases at STP occupies close to the ideal molar volume.


Figure 10.10 Comparison of molar volumes at STP.

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Relating the Ideal-Gas Equation and the Gas Laws

• We can start with the ideal-gas equation and deriverelationships between any other two variables, V and T(Charles’s law), n and V (Avogadro’s law), or P and T.– If n and T are constant, the values of P and V can change, but

the product PV must remain constant.

– If P, V, and T all change for a fixed number of moles of gas, the ideal-gas equation gives

– This equation is often called the combined gas law.


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Gas Densities and Molar Mass

• Calculation of the density (d = m/V) and the molar mass of a gas using the ideal-gas equation.– We can arrange the ideal-gas equation to

– If we multiply both sides of this equation by the molar mass, M (the number of grams in 1 mol of a substance), we obtain

– The term on the left equals the density in grams per liter:

– Thus, the density of the gas is also given by

– We can rearrange the equation to solve for the molar mass of a gas:

10.5 FURTHER APPLICATIONS OF THE IDEAL-GAS EQUATION

n × M = m

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Gas Mixtures and Partial Pressures

• Dalton’s law of partial pressures: The total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone.– Partial pressure: The pressure exerted by a particular component

of a mixture of gases.

– If each gas obeys the ideal-gas equation, we can write

– All the gases are at the same temperature and occupy the same volume. Therefore, we obtain

10.6 GAS MIXTURES AND PARTIAL PRESSURES

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Partial Pressures and Mole Fractions

• Because each gas in a mixture behaves independently, the amount of a given gas in a mixture can be related to its partial pressure.– For an ideal gas, we can write

– The ratio n1/nt is called the mole fraction of gas 1, which we denote X1.– The mole fraction (X): A dimensionless number that expresses the

ratio of the number of moles of one component in a mixture to the total number of moles in the mixture.

– Thus, for gas 1 we have


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Partial Pressures and Mole Fractions

• Ex) N2 molecules in air.– The mole fraction of N2 in air is 0.78.– 78% of the molecules in air are N2.– If the barometric pressure is 760 torr, the partial pressure of N2 is

– Because N2 makes up 78% of the mixture, it contributes 78% of the total pressure.


Air is approximately 78% nitrogen, 21% oxygen, plus a mixture of many other gases.

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The Kinetic-Molecular Theory

• This is a model that aids in our understanding of what happens to gas particles as environmental conditions change.

• Summaries of the kinetic-molecular theory.

10.7 THE KINETIC-MOLECULAR THEORY OF GASES

Figure 10.12 The molecular origin of gas pressure.

1. Gases consist of large numbers of molecules that are in continuous, random motion.

2. The combined volume of all the molecules of the gas is negligiblerelative to the total volume in which the gas is contained.

3. Attractive and repulsive forces between gas molecules are negligible.

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The Kinetic-Molecular Theory

4. Energy can be transferred between molecules during collisions but, as long as temperature remains constant, the average kinetic energy of the molecules does not change with time.

5. The average kinetic energy of the molecules is proportional to the absolute temperature. At any given temperature the molecules of all gases have the same average kinetic energy.


Figure 10.12 The molecular origin of gas pressure.

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Distributions of Molecular Speed

• Although collectively the molecules in a sample of gas have an average kinetic energy and hence an average speed, the individual molecules are moving at different speeds.

• The average kinetic energy of the molecules is proportional to the absolute temperature.


Figure 10.13 Distribution of molecular speeds for nitrogen gas. (a) The effect of temperature on molecular speed. The relative area under the curve for a range of speeds gives the relative fraction of molecules that have those speed. (b) Position of most probable (ump), average(uav), and root-mean-square (urms) speeds of gas molecules. The data shown here are for nitrogen gas at 0 °C.

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Distributions of Molecular Speed

• Three different molecular speeds:– ump is the most probable speed (most molecules are this fast).– uav is the average speed of the molecules.– urms, the root-mean-square speed, is the one associated with their

average kinetic energy (½m(urms)2).


Figure 10.13 Distribution of molecular speeds for nitrogen gas. (a) The effect of temperature on molecular speed. The relative area under the curve for a range of speeds gives the relative fraction of molecules that have those speed. (b) Position of most probable (ump), average(uav), and root-mean-square (urms) speeds of gas molecules. The data shown here are for nitrogen gas at 0 °C.

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Application of Kinetic-Molecular Theoryto the Gas Laws

1. An increase in volume at constant temperature causes pressure to decrease.– When the volume is increased, the molecules must move a longer

distance between collisions. Consequently, there are fewer collisions per unit time with the container walls, which means the pressure decreases.

2. A temperature increase at constant volume causes pressure to increase.– An increase in temperature means an increase in the average kinetic

energy of the molecules and in urms. The temperature increase causes more collisions with the walls per unit time because the molecules are all moving faster. A greater number of more forceful collisions means the pressure increases, and the theory explains this increase.


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Molecular Effusion and Diffusion

• Using the assumptions of kinetic-molecular theory, we can show that:

P = Pressure, N = Number of molecules, V = Volume, m = Mass of each molecule, u2 = Average of the squares of the speeds of the molecules.

• We can derive the equation:

M = Molar mass of the molecules.– The less massive the gas molecules, the higher their rms speed.

10.8 MOLECULAR EFFUSION AND DIFFUSION

3RTurms = u2 = ——

M

1 NP = — · — · m · u2

3 V

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• The particles of the lighter gas must have a higher rms speed than the particles of the heavier one.

• The most probable speed of a gas molecule can also be derived:


Figure 10.14 The effect of molar mass on molecular speed at 25 °C.

3RTurms = ——

M

2RTump = ——

M

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• The dependence of molecular speed on mass has two interesting consequences, effusion and diffusion.

• Effusion: The escape of gas molecules through a tiny hole into an evacuated space.

• Diffusion: the spread of one substance throughout a space or throughout a second substance.– Ex) The molecules of a perfume

diffuse throughout a room.


Figure 10.15 Effusion.

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Graham’s Law of Effusion

• Graham’s law: The effusion rate of a gas is inversely proportional to the square root of its molar mass.– If the rates of effusion of the two gases are r1 and r2 and their

molar masses are M1 and M2,

– The lighter gas has the higher effusion rate.– The rate of effusion is directly proportional to the rms speed of

the molecules.


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Graham’s Law of Effusion

– Ex) The difference in the rates of effusion for helium and argon explains why a helium balloon would deflate faster.


Figure 10.16 An illustration of Graham’s law of effusion.

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Diffusion and Mean Free Path

• The diffusion of gases is much slower than molecular speeds because of molecular collisions.

• The molecule’s mean free path: The average distance traveled by the molecule between collisions.


Figure 10.18 Diffusion of a gas molecule. For clarity, no other gas molecules in the container are shown.

• The mean free path for air molecules is about 60 nm at sea level and about 10 cm at ~100 km in altitude (much lower air density).

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Real Gases: Deviations from Ideal Behavior

• In the real world, the behavior of gases only conforms to the ideal-gas equation at relatively high temperature and low pressure.

10.9 REAL GASES: DEVIATIONS FROM IDEAL BEHAVIOR

Figure 10.19 The effect of pressure on the behavior of several real gases. Data for 1 mol of gas in all cases. Data for N2, CH4, and H2 are at 300 K; for CO2 data are at 313 K because under high pressure CO2 liquefies at 300 K.

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• Even the same gas will show wildly different behavior under high pressure at different temperatures.


Figure 10.20 The effect of temperature and pressure on the behavior of nitrogen gas.

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• The assumptions made in the kinetic-molecular model (negligible volume of gas molecules themselves, no attractive forces between gas molecules, etc.) breakdown at high pressure and/or low temperature.


Figure 10.21 Gases behave more ideally at low pressure than at high pressure.

Figure 10.22 In any real gas, attractive intermolecular forces reduce pressure to values lower than in an ideal gas.

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Real Gases: Deviations from Ideal Behavior10.9 REAL GASES: DEVIATIONS FROM IDEAL BEHAVIOR

At high P, gas volumesare not negligible.

Figure 10.19 The effect of pressure on the behavior of several real gases. Data for 1 mol of gas in all cases. Data for N2, CH4, and H2 are at 300 K; for CO2 data are at 313 K because under high pressure CO2 liquefies at 300 K.

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Real Gases: Deviations from Ideal Behavior10.9 REAL GASES: DEVIATIONS FROM IDEAL BEHAVIOR

Cooling a gas increase the chance for molecules to interact with each other.

Attractive forces between molecules reduce the pressure.

Figure 10.20 The effect of temperature and pressure on the behavior of nitrogen gas.

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The van der Waals Equation

• The ideal-gas equation can be adjusted to takethese deviations from ideal behavior into account.

• The corrected ideal-gas equation is known as the van der Waals equation.

– The term n2a/V2 accounts for the attractive forces.– The term nb accounts for the small but finite volume

occupied by the gas molecules.


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The van der Waals Equation

– a and b values increase with mass of the molecule and the complexity of its structure.


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Chapter 10. HomeworkExercises 10.9

10.2310.2510.3710.4310.5210.5710.6810.7410.7610.8110.9510.103

chapter 10: gases - sogang

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