《物理化学》课程参考书籍：《ATKINS' Physical Chemistry》PDF电子书英文原版，Eighth Edition，2006（Peter Atkins、Julio de Paula）.pdf_P31-P35

6 1 THE PROPERTIES OF GASES metal container has diathermic walls.a boundary is adiabatic (thermally insulating if no change occurs even though the two objects have different temperatures.A vacuum flask is an approximation to an adiabatic container. The temperature is a property that indicates whether two objects would be in ey were in contact through a di thermic uilibrium is csta oundary.Thermal n)is in the ith an obiect B(a block of that bis alsc n the rmal eq uilibrium with t C(a flask of war The has been found experimentally that A and C will also be in thermal equilibrium when they are put in contact(Fig.1.3).This observation is summarized by the Zeroth Law of thermodynamics: with and Bis in thermal equilibrium withC then 13The experience summarized by the Band Bisin thermal The Zeroth Law just t of ter use ofa ther a device for meas e that bis a glass canillary con then C is in thermal equilibrium with A. taining a liquid,such as mercury,that expands significantly as the temperature increases.Then,when A is in contact with B,the mercury column in the latter has a certain length.According to the Zeroth Law,if the mercury column in B has the same te o will occur whe hgyareinthofthctempcratarc of the mercury column as a measure o In the carly days of thermo netry (and still in laboratory nractice todav)ter tures were related to the length of a column of liquid,and the difference in length shown when the thermometer was first in contact with melting ice and then with boiling water was divided into 100 steps called'degrees',the lower point being labelled Ihis procedure led to the Celsius scale of temperature.In this text,tempera on the C xpresse d in degree cause erent liqui expand to a d de showed different n ature hetu n their fived noints The pressure of a gas,how er can he used to construct a perfect-gas temperature scale that is independent of the identity of the gas.The perfect-gas scale turns out to be identical to the thermodynamic temperature scale to be introduced in Section 3.2c oe T/K=0/C+273.15 (1.4 This relation,in the form C=T/K-273.15,is the current definition of the Celsius scale in terms of the more fundamental Kelvin scale.It implies that a difference in temperature of 1C is equivalent to a difference of I K on the thermodynamic temperature scale reis regardless of the si o tedoon the scua ure. n as bar o the Celsius

6 1 THE PROPERTIES OF GASES metal container has diathermic walls. A boundary is adiabatic (thermally insulating) if no change occurs even though the two objects have different temperatures. A vacuum flask is an approximation to an adiabatic container. The temperature is a property that indicates whether two objects would be in ‘thermal equilibrium’ if they were in contact through a diathermic boundary. Thermal equilibrium is established if no change of state occurs when two objects A to B are in contact through a diathermic boundary. Suppose an object A (which we can think of as a block of iron) is in thermal equilibrium with an object B (a block of copper), and that B is also in thermal equilibrium with another object C (a flask of water). Then it has been found experimentally that A and C will also be in thermal equilibrium when they are put in contact (Fig. 1.3). This observation is summarized by the Zeroth Law of thermodynamics: If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C is also in thermal equilibrium with A. The Zeroth Law justifies the concept of temperature and the use of a thermometer, a device for measuring the temperature. Thus, suppose that B is a glass capillary containing a liquid, such as mercury, that expands significantly as the temperature increases. Then, when A is in contact with B, the mercury column in the latter has a certain length. According to the Zeroth Law, if the mercury column in B has the same length when it is placed in thermal contact with another object C, then we can predict that no change of state of A and C will occur when they are in thermal contact. Moreover, we can use the length of the mercury column as a measure of the temperatures of A and C. In the early days of thermometry (and still in laboratory practice today), temperatures were related to the length of a column of liquid, and the difference in lengths shown when the thermometer was first in contact with melting ice and then with boiling water was divided into 100 steps called ‘degrees’, the lower point being labelled 0. This procedure led to the Celsius scale of temperature. In this text, temperatures on the Celsius scale are denoted θ and expressed in degrees Celsius (°C). However, because different liquids expand to different extents, and do not always expand uniformly over a given range, thermometers constructed from different materials showed different numerical values of the temperature between their fixed points. The pressure of a gas, however, can be used to construct a perfect-gas temperature scale that is independent of the identity of the gas. The perfect-gas scale turns out to be identical to the thermodynamic temperature scale to be introduced in Section 3.2c, so we shall use the latter term from now on to avoid a proliferation of names. On the thermodynamic temperature scale, temperatures are denoted T and are normally reported in kelvins, K (not °K). Thermodynamic and Celsius temperatures are related by the exact expression T/K = θ/°C + 273.15 (1.4) This relation, in the form θ/°C = T/K − 273.15, is the current definition of the Celsius scale in terms of the more fundamental Kelvin scale. It implies that a difference in temperature of 1°C is equivalent to a difference of 1 K. A note on good practice We write T = 0, not T = 0 K for the zero temperature on the thermodynamic temperature scale. This scale is absolute, and the lowest temperature is 0 regardless of the size of the divisions on the scale (just as we write p = 0 for zero pressure, regardless of the size of the units we adopt, such as bar or pascal). However, we write 0°C because the Celsius scale is not absolute. B A C Equilibrium Equilibrium Equilibrium Fig. 1.3 The experience summarized by the Zeroth Law of thermodynamics is that, if an object A is in thermal equilibrium with B and B is in thermal equilibrium with C, then C is in thermal equilibrium with A

1.2 THE GAS LAWS 7 lllustration 1.1 Converting temperatures To express 25.00Cas a temperature in kelvins,we useeqn 1.to write T7K=(25.00C)/℃+273.15=25.00+273.15=298.15 Note how the units (in this case,C)are cancelled like numbers.This is the proced. ure called'quantity calculus'in which a physical quantity (such as the temperature) ncreasing is the product ofa numerical value(25.00)anda unit(1C).Multiplication ofboth sides by the unit K then gives T=298.15 K. A note on good practice When the units need to be specified in an equation,the approved procedure,which avoids any ambiguity,is to write (physical quantity) units,which is a dimensionless number,just as(25.00C)/C=25.00 in this Illustration.Units may be multiplied and cancelled just like numbers. 1.2 The gas laws Volume,V The equation of state of a gas at low pressure was established by combining a series of empirical laws. 间The perfect gas law We assume that the following individual gas laws are familiar: Boyle'slaw:pV=constant,at constantn,T (1.5)9 Charles'slaw:V=constant xT,at (1.6a)° p=constant x T,at constant V (1.6b)9 Avogadro's principle:2V=constant x nat constant p,T (1.7° Boyle's and Charles's laws are examples ofa limitinglaw,alaw that is strictly true only in a certain limit,in this case p0.Equations valid in this limiting sense will be signalled by aon the equation number,as in these expressions.Avogadro's principle is commonly expre in the form 'equal volume of gases at the same temperature ne num bers of mo In this onably reliable at n chemistrv. Figure 1.4 depicts the variation of the pressure of a sample of gas as the volume is changed.Each of the curves in the graph corresponds to a single temperature and hence is called an isotherm.According to Boyle's law,the isotherms of gases are An alterna tive depiction,a plot of pressure against 1/vo um is sho The lir mperature summari arles Comment 1.2 16 on are of isob A hyperbola is a curve obtained by 17 plotting y against x with xy constant or lines showing the variation of properties at constant volume. text's web site

1.2 THE GAS LAWS 7 2 Avogadro’s principle is a principle rather than a law (a summary of experience) because it depends on the validity of a model, in this case the existence of molecules. Despite there now being no doubt about the existence of molecules, it is still a model-based principle rather than a law. 3 To solve this and other Explorations, use either mathematical software or the Living graphs from the text’s web site. Comment 1.2 A hyperbola is a curve obtained by plotting y against x with xy = constant. Volume, V Pressure, p Increasing temperature, T 0 0 Fig. 1.4 The pressure–volume dependence of a fixed amount of perfect gas at different temperatures. Each curve is a hyperbola (pV = constant) and is called an isotherm. Exploration3 Explore how the pressure of 1.5 mol CO2(g) varies with volume as it is compressed at (a) 273 K, (b) 373 K from 30 dm3 to 15 dm3 . Illustration 1.1 Converting temperatures To express 25.00°C as a temperature in kelvins, we use eqn 1.4 to write T/K = (25.00°C)/°C + 273.15 = 25.00 + 273.15 = 298.15 Note how the units (in this case, °C) are cancelled like numbers. This is the procedure called ‘quantity calculus’ in which a physical quantity (such as the temperature) is the product of a numerical value (25.00) and a unit (1°C). Multiplication of both sides by the unit K then gives T = 298.15 K. A note on good practice When the units need to be specified in an equation, the approved procedure, which avoids any ambiguity, is to write (physical quantity)/ units, which is a dimensionless number, just as (25.00°C)/°C = 25.00 in this Illustration. Units may be multiplied and cancelled just like numbers. 1.2 The gas laws The equation of state of a gas at low pressure was established by combining a series of empirical laws. (a) The perfect gas law We assume that the following individual gas laws are familiar: Boyle’s law: pV = constant, at constant n, T (1.5)° Charles’s law: V = constant × T, at constant n, p (1.6a)° p = constant × T, at constant n, V (1.6b)° Avogadro’s principle:2 V = constant × n at constant p, T (1.7)° Boyle’s and Charles’s laws are examples of a limiting law, a law that is strictly true only in a certain limit, in this case p → 0. Equations valid in this limiting sense will be signalled by a ° on the equation number, as in these expressions. Avogadro’s principle is commonly expressed in the form ‘equal volumes of gases at the same temperature and pressure contain the same numbers of molecules’. In this form, it is increasingly true as p → 0. Although these relations are strictly true only at p = 0, they are reasonably reliable at normal pressures (p ≈ 1 bar) and are used widely throughout chemistry. Figure 1.4 depicts the variation of the pressure of a sample of gas as the volume is changed. Each of the curves in the graph corresponds to a single temperature and hence is called an isotherm. According to Boyle’s law, the isotherms of gases are hyperbolas. An alternative depiction, a plot of pressure against 1/volume, is shown in Fig. 1.5. The linear variation of volume with temperature summarized by Charles’s law is illustrated in Fig. 1.6. The lines in this illustration are examples of isobars, or lines showing the variation of properties at constant pressure. Figure 1.7 illustrates the linear variation of pressure with temperature. The lines in this diagram are isochores, or lines showing the variation of properties at constant volume

8 1 THE PROPERTIES OF GASES A note on good practice To test the validity of a relation between two quantities, it is best to plot them in such a way that they should give a straight line, for deviations from a straight line are much easier to detect than deviations from a curve. The empirical observations summarized by eqns 1.5–7 can be combined into a single expression: pV = constant × nT This expression is consistent with Boyle’s law (pV = constant) when n and T are constant, with both forms of Charles’s law (p ∝ T, V ∝ T) when n and either V or p are held constant, and with Avogadro’s principle (V ∝ n) when p and T are constant. The constant of proportionality, which is found experimentally to be the same for all gases, is denoted R and called the gas constant. The resulting expression pV = nRT (1.8)° is the perfect gas equation. It is the approximate equation of state of any gas, and becomes increasingly exact as the pressure of the gas approaches zero. A gas that obeys eqn 1.8 exactly under all conditions is called a perfect gas (or ideal gas). A real gas, an actual gas, behaves more like a perfect gas the lower the pressure, and is described exactly by eqn 1.8 in the limit of p → 0. The gas constant R can be determined by evaluating R = pV/nT for a gas in the limit of zero pressure (to guarantee that it is 1/V Pressure, p 0 0 Increasing temperature, T Extrapolation Volume, V 0 Decreasing pressure, p Extrapolation Temperature,T 0 Pressure, p 0 0 Extrapolation Decreasing volume, V Temperature, T Fig. 1.5 Straight lines are obtained when the pressure is plotted against 1/V at constant temperature. Exploration Repeat Exploration 1.4, but plot the data as p against 1/V. Fig. 1.6 The variation of the volume of a fixed amount of gas with the temperature at constant pressure. Note that in each case the isobars extrapolate to zero volume at T = 0, or θ = −273°C. Exploration Explore how the volume of 1.5 mol CO2(g) in a container maintained at (a) 1.00 bar, (b) 0.50 bar varies with temperature as it is cooled from 373 K to 273 K. Fig. 1.7 The pressure also varies linearly with the temperature at constant volume, and extrapolates to zero at T = 0 (−273°C). Exploration Explore how the pressure of 1.5 mol CO2(g) in a container of volume (a) 30 dm3 , (b) 15 dm3 varies with temperature as it is cooled from 373 K to 273 K

1.2 THE GAS LAWS 9 Table 1.2 The gas constant R 8.314 47 J K−1 mol−1 8.205 74 × 10−2 dm3 atm K−1 mol−1 8.314 47 × 10−2 dm3 bar K−1 mol−1 8.314 47 Pa m3 K−1 mol−1 1 62.364 dm3 Torr K−1 mol−1 1.987 21 cal K−1 mol−1 Comment 1.3 For an object of mass m moving at a speed 1, the kinetic energy is EK = 1 –2m12 . The potential energy, EP or V, of an object is the energy arising from its position (not speed). No universal expression for the potential energy can be given because it depends on the type of interaction the object experiences. behaving perfectly). However, a more accurate value can be obtained by measuring the speed of sound in a low-pressure gas (argon is used in practice) and extrapolating its value to zero pressure. Table 1.2 lists the values of R in a variety of units. Molecular interpretation 1.1 The kinetic model of gases The molecular explanation of Boyle’s law is that, if a sample of gas is compressed to half its volume, then twice as many molecules strike the walls in a given period of time than before it was compressed. As a result, the average force exerted on the walls is doubled. Hence, when the volume is halved the pressure of the gas is doubled, and p × V is a constant. Boyle’s law applies to all gases regardless of their chemical identity (provided the pressure is low) because at low pressures the average separation of molecules is so great that they exert no influence on one another and hence travel independently. The molecular explanation of Charles’s law lies in the fact that raising the temperature of a gas increases the average speed of its molecules. The molecules collide with the walls more frequently and with greater impact. Therefore they exert a greater pressure on the walls of the container. These qualitative concepts are expressed quantitatively in terms of the kinetic model of gases, which is described more fully in Chapter 21. Briefly, the kinetic model is based on three assumptions: 1. The gas consists of molecules of mass m in ceaseless random motion. 2. The size of the molecules is negligible, in the sense that their diameters are much smaller than the average distance travelled between collisions. 3. The molecules interact only through brief, infrequent, and elastic collisions. An elastic collision is a collision in which the total translational kinetic energy of the molecules is conserved. From the very economical assumptions of the kinetic model, it can be deduced (as we shall show in detail in Chapter 21) that the pressure and volume of the gas are related by pV = nMc 2 (1.9)° where M = mNA, the molar mass of the molecules, and c is the root mean square speed of the molecules, the square root of the mean of the squares of the speeds, v, of the molecules: c = v2 1/2 (1.10) We see that, if the root mean square speed of the molecules depends only on the temperature, then at constant temperature pV = constant which is the content of Boyle’s law. Moreover, for eqn 1.9 to be the equation of state of a perfect gas, its right-hand side must be equal to nRT. It follows that the root mean square speed of the molecules in a gas at a temperature T must be (1.11)° We can conclude that the root mean square speed of the molecules of a gas is proportional to the square root of the temperature and inversely proportional to the square root of the molar mass. That is, the higher the temperature, the higher the root mean square speed of the molecules, and, at a given temperature, heavy molecules travel more slowly than light molecules. The root mean square speed of N2 molecules, for instance, is found from eqn 1.11 to be 515 m s−1 at 298 K. c RT M / = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3 1 2 1 3