Topic 1A The Perfect Gas PDF

# TOPIC 1A The Perfect Gas ## Why Do You Need to Know This Material? The relation between the pressure, volume, and temperature of a perfect gas is used extensively in the development of quantitative theories about the physical and chemical behavior of real gases. It is also used extensively throug...

# TOPIC 1A The Perfect Gas ## Why Do You Need to Know This Material? The relation between the pressure, volume, and temperature of a perfect gas is used extensively in the development of quantitative theories about the physical and chemical behavior of real gases. It is also used extensively throughout thermodynamics. ## What Is the Key Idea? The perfect gas law, which describes the relation between the pressure, volume, temperature, and amount of substance, is a limiting law that is obeyed increasingly well as the pressure of a gas tends to zero. ## What Do You Need to Know Already? You need to know how to handle quantities and units in calculations, as reviewed in the Resource section. The properties of gases were among the first to be established quantitatively (largely during the seventeenth and eighteenth centuries) when the technological requirements of travel in balloons stimulated their investigation. This Topic reviews how the physical state of a gas is described using variables such as pressure and temperature, and then discusses how these variables are related. ## 1A.1 Variables of State The physical state of a sample of a substance, its physical condition, is defined by its physical properties. Two samples of the same substance that have the same physical properties are said to be in the same state. The variables of state, the variables needed to specify the state of a system, are the amount of substance it contains, *n*; the volume it occupies, *V*; the pressure, *p*; and the temperature, *T*. ### 1A.1(a) Pressure and Volume The pressure, *p*, that an object experiences is defined as the force, *F*, applied divided by the area, *A*, to which that force is applied. A gas exerts a pressure on the walls of its container as a result of the collisions between the molecules and the walls: these collisions are so numerous that the force, and hence the pressure, is steady. The SI unit of pressure is the pascal, Pa, defined as 1 Pa = 1 Nm¯² = 1 kgm˜¯¹s¯². Several other units are still widely used, and the relations between them are given in Table 1A.1. Because many physical properties depend on the pressure acting on a sample, it is appropriate to select a certain value of the pressure to report their values. The standard pressure, *p₀*, for reporting physical quantities is currently defined as *p* = 1 bar (that is, 10⁵ Pa) exactly. This pressure is close to, but not the same as, 1 atm, which is typical for everyday conditions. Consider the arrangement shown in Fig. 1A.1 where two gases in separate containers share a common movable wall. In Fig. 1A.1a the gas on the left is at higher pressure than that on the right, and so the force exerted on the wall by the gas on the left is greater than that exerted by the gas on the right. As a result, the wall moves to the right, the pressure on the left decreases, and that on the right increases. Eventually (as in Fig. 1A.1b) the two pressures become equal and the wall no longer moves. This condition of equality of pressure on either side of a movable wall is a state of mechanical equilibrium between the two gases. The pressure exerted by the atmosphere is measured with a barometer. The original version of a barometer (which was invented by Torricelli, a student of Galileo) involved taking a glass tube, sealed at one end, filling it with mercury and then up-ending it (without letting in any air) into a bath of mercury. The pressure of the atmosphere acting on the surface of the mercury in the bath supports a column of mercury of a certain height in the tube: the pressure at the base of the column, due to the mercury in the tube, is equal to the atmospheric pressure. As the atmospheric pressure changes, so does the height of the column. The pressure of gas in a container, and also now the atmosphere, is measured by using a pressure gauge, which is a device with properties that respond to pressure. For instance, in a Bayard-Alpert pressure gauge the molecules present in the gas are ionized and the resulting current of ions is interpreted in terms of the pressure. In a capacitance manometer, two electrodes form a capacitor. One electrode is fixed and the other is a diaphragm which deflects as the pressure changes. This deflection causes a change in the capacitance, which is measured and interpreted as a pressure. Certain semiconductors also respond to pressure and are used as transducers in solid-state pressure gauges, including those in mobile phones (cell phones). The volume, *V*, of a gas is a measure of the extent of the region of space it occupies. The SI unit of volume is m³. ### 1A.1(b) Temperature The temperature is formally a property that determines in which direction energy will flow as heat when two samples are placed in contact through thermally conducting walls: energy flows from the sample with the higher temperature to the sample with the lower temperature. The symbol *T* denotes the thermodynamic temperature, which is an absolute scale with *T*=0 as the lowest point. Temperatures above *T* = 0 are expressed by using the Kelvin scale, in which the gradations of temperature are expressed in kelvins (K; not °K). Until 2019, the Kelvin scale was defined by setting the triple point of water (the temperature at which ice, liquid water, and water vapour are in mutual equilibrium) at exactly 273.16 K. The scale has now been redefined by referring it to the more precisely known value of the Boltzmann constant. There are many devices used to measure temperature. They vary from simple devices that measure the expansion of a liquid along a tube, as commonly found in laboratories, to electronic devices where the resistance of a material or the potential difference developed at a junction is related to the temperature. The Celsius scale of temperature is commonly used to express temperatures. In this text, temperatures on the Celsius scale are denoted θ (theta) and expressed in degrees Celsius (°C). The thermodynamic and Celsius temperatures are related by the exact expression: $T/K = 0/°C + 273.15$ This relation is the 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. The lowest temperature on the thermodynamic temperature scale is written *T* = 0, not *T*=0K. This scale is absolute, and the lowest temperature is 0 regardless of the size of the divisions on the scale (just as zero pressure is denoted *p* = 0, regardless of the size of the units, such as bar or pascal). However, it is appropriate to write 0 °C because the Celsius scale is not absolute. ### 1A.1(c) Amount In day-to-day conversation 'amount' has many meanings but in physical science it has a very precise definition. The amount of substance, *n*, is a measure of the number of specified entities present in the sample; these entities may be atoms, or molecules, or formula units. The SI unit of amount of substance is the mole (mol). The amount of substance is commonly referred to as the 'chemical amount' or simply 'amount'. Until 2019 the mole was defined as the number of carbon atoms in exactly 12 g of carbon-12. However, it has been redefined such that 1 mol of a substance contains exactly 6.02214076×10²³ entities. The number of entities per mole is called Avogadro's constant, *N₁*. It follows from the definition of the mole that *N₁* =6.02214076×10²³ mol¯¹. Note that *N₁* is a constant with units, not a pure number. Also, it is not correct to specify amount as the 'number of moles': the correct phrase is 'amount in moles. The amount of substance is related to the mass, *m*, of the substance through the molar mass, *M*, which is the mass per mole of its atoms, its molecules, or its formula units. The SI unit of molar mass is kg mol¹ but it is more common to use gmol¹. The amount of substance of specified entities in a sample can readily be calculated from its mass by using: $n = \frac{m}{M}$ ### 1A.1(d) Intensive and Extensive Properties Suppose a sample is divided into smaller samples. If a property of the original sample has a value that is equal to the sum of its values in all the smaller samples, then it is said to be an extensive property. Amount, mass, and volume are examples of extensive properties. If a property retains the same value as in the original sample for all the smaller samples, then it is said to be intensive. Temperature and pressure are examples of intensive properties. The value of a property *X* divided by the amount *n* gives the molar value of that property *X*: that is, *X* = *X*/n. All molar properties are intensive, whereas *X* and *n* are both extensive. The mass density, *p* = *m*/V, is also intensive. ## 1A.2 Equations of State Although in principle the state of a pure substance is specified by giving the values of *n*, *V*, *p*, and *T*, it has been established experimentally that it is sufficient to specify only three of these variables because doing so fixes the value of the fourth variable. That is, it is an experimental fact that each substance is described by an equation of state, an equation that interrelates these four variables. The general form of an equation of state is: $p = f(T,V,n)$ This equation means that if the values of *n*, *T*, and *V* are known for a particular substance, then the pressure has a fixed value. Each substance is described by its own equation of state, but the explicit form of the equation is known in only a few special cases. One very important example is the equation of state of a 'perfect gas, which has the form *p* = *nRT*/V, where *R* is a constant independent of the identity of the gas. ### 1A.2(a) The Empirical Basis of the Perfect Gas Law The equation of state of a perfect gas was established by combining a series of empirical laws that arose from experimental observations. These laws can be summarized as: * **Boyle's law:** *pV* = constant, at constant *n*,*T* * **Charles's law:** *V* = constant × *T*, at constant *n*, *p* * **Charles's law:** *p* = constant×*T*, at constant *n*, *V* * **Avogadro's principle:** *V* = constant×*n*, at constant *p*,*T* Boyle's and Charles's laws are strictly true only in the limit that the pressure goes to zero (*p* → 0): they are examples of a limiting law, a law that is strictly true only in a certain limit. However, these laws are found to be reasonably reliable at normal pressures (*p*≈ 1 bar) and are used throughout chemistry. Avogadro's principle is so-called because it supposes that the system consists of molecules whereas a law is strictly a summary of observations and independent of any assumed model. Figure 1A.2 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 (curves obtained by plotting *y* against *x* with *xy* = constant, or *y* = constant/*x*). An alternative depiction, a plot of pressure against 1/volume, is shown in Fig. 1A.3; in such a plot the isotherms are straight lines because *p* is proportional to 1/V. Note that all the lines extrapolate to the point *p* = 0, 1/V = 0 but have slopes that depend on the temperature. The linear variation of volume with temperature summarized by Charles's law is illustrated in Fig. 1A.4. The lines in this illustration are examples of isobars, or lines showing the variation of properties at constant pressure. All these isobars extrapolate to the point *V* = 0, *T* = 0 and have slopes that depend on the pressure. Figure 1A.5 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, and they all extrapolate to *p* = 0, *T* = 0. The empirical observations summarized by Boyle's and Charles's laws and Avogadro's principle 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. It is also consistent with both forms of Charles's law: *p*∝*T* when *n* and *V* are held constant, and *V*∝*T* when *n* and *p* are held constant. The expression also agrees 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 (molar) gas constant. The resulting expression: *PV* = *nRT* is the perfect gas law (or perfect gas equation of state). A gas that obeys this law exactly under all conditions is called a perfect gas (or ideal gas). Although the term 'ideal gas' is used widely, in this text we prefer to use 'perfect gas' because there is an important and useful distinction between ideal and perfect. The distinction is that in an ideal system' all the interactions between molecules are the same; in a 'perfect system, not only are they the same but they are also zero. For a real gas, any actual gas, the perfect gas law is approximate, but the approximation becomes better as the pressure of the gas approaches zero. In the limit that the pressure goes to zero, *p*→ 0, the equation is exact. The value of 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 behaving perfectly). As remarked in Energy: A first look, the modern procedure is to note that *R* = *N₁*k, where *k* is Boltzmann's constant and *N₁* has its newly defined value, as indicated earlier. The surface in Fig. 1A.6 is a plot of the pressure of a fixed amount of perfect gas molecules against its volume and thermodynamic temperature as given by eqn 1A.4. The surface depicts the only possible states of a perfect gas: the gas cannot exist in states that do not correspond to points on the surface. Figure 1A.7 shows how the graphs in Figs. 1A.2, 1A.4, and 1A.5 correspond to sections through the surface. ## Example 1A.2 Using the Perfect Gas Law Nitrogen gas is introduced into a vessel of constant volume at a pressure of 100 atm and a temperature of 300 K. The temperature is then raised to 500 K. What pressure would the gas then exert, assuming that it behaved as a perfect gas? **Collect your thoughts** The pressure is expected to be greater on account of the increase in temperature. The perfect gas law in the form *pV*/nT = *R* implies that if the conditions are changed from one set of values to another, then because *pV*/nT is equal to a constant, the two sets of values are related by the 'combined gas law': $\frac{p_1 V_1 }{n_1 T_1} = \frac{p_2 V_2 }{n_2 T_2}$ In this case the volume is the same before and after heating, so *V₁* = *V₂* and these terms cancel. Likewise the amount does not change upon heating, so *n₁* = *n₂* and these terms also cancel. **The Solution** Cancellation of the volumes and amounts on each side of the combined gas law results in: $\frac{p_1}{T_1} = \frac{p_2}{T_2}$ which can be rearranged into: $p_2 = p_1 \frac{T_2}{T_1}$ Substitution of the data then gives $p_2 = \frac{500 K}{300 K} × (100 atm) = 167 atm$ 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 *pV* 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. For a quantitative account of these relations, see Topic 1B. ### 1A.2(b) The Value of the Gas Constant If the pressure, volume, amount, and temperature are expressed in their SI units the gas constant *R* has units Nm K¯¹ mol¯¹ which, because 1 J = 1 N m, can be expressed in terms of JK¯¹ mol¯¹. The currently accepted value of *R* is 8.3145 JK¯¹ mol¯¹. Other combinations of units for pressure and volume result in different values and units for the gas constant. Some commonly encountered combinations are given in Table 1A.2. The perfect gas law is of the greatest importance in physical chemistry because it is used to derive a wide range of relations found throughout thermodynamics. It is also of considerable practical utility for calculating the properties of a perfect gas under a variety of conditions. For instance, the molar volume, *V* = *V*/n, of a perfect gas under the conditions called standard ambient temperature and pressure (SATP), defined as 298.15 K and 1 bar, is calculated as 24.789 dm³ mol¹. An earlier definition, standard temperature and pressure (STP), was 0 °C and 1 atm; at STP, the molar volume of a perfect gas under these conditions is 22.414 dm³ mol¯¹. ### 1A.2(c) Mixtures of Gases When dealing with gaseous mixtures, it is often necessary to know the contribution that each component makes to the total pressure of the sample. The partial pressure, *P₁*, of a gas *J* in a mixture (any gas, not just a perfect gas), is defined as: *P₁* = *x₁*P where *x₁* is the mole fraction of the component *J*, the amount of *J* expressed as a fraction of the total amount of molecules, *n*, in the sample: *x₁* = *n₁*/n When no *J* molecules are present, *x₁* = 0; when only *J* molecules are present, *x₁* = 1. It follows from the definition of *x₁* that, whatever the composition of the mixture, *x₁* + *x₂* + ... = 1 and therefore that the sum of the partial pressures is equal to the total pressure: *P₁* + *P₂* +... = (*x₁* + *x₂* +...)*p* = *p* This relation is true for both real and perfect gases. When all the gases are perfect, the partial pressure as defined in eqn 1A.6 is also the pressure that each gas would exert if it occupied the same container alone at the same temperature. The latter is the original meaning of 'partial pressure'. That identification was the basis of the original formulation of Dalton's law: The pressure exerted by a mixture of gases is the sum of the pressures that each one would exert if it occupied the container alone. This law is valid only for mixtures of perfect gases, so it is not used to define partial pressure. Partial pressure is defined by eqn 1A.6, which is valid for all gases. ## Example 1A.3 Calculating Partial Pressures The mass percentage composition of dry air at sea level is approximately N₂: 75.5; O₂: 23.2; Ar: 1.3. What is the partial pressure of each component when the total pressure is 1.20 atm? **Collect your thoughts** Partial pressures are defined by eqn 1A.6. To use the equation, first calculate the mole fractions of the components by using eqn 1A.7 and the fact that the amount of atoms or molecules *J* of molar mass *M*, in a sample of mass *m*, is *n₁* = *m*/M. The mole fractions are independent of the total mass of the sample, so choose the latter to be exactly 100g (which makes the conversion from mass percentages very straightforward). Thus, the mass of N₂ present is 75.5 per cent of 100g, which is 75.5 g. **The Solution** The amounts of each type of atom or molecule present in 100 g of air, in which the masses of N₂, O₂, and Ar are 75.5 g, 23.2 g, and 1.3 g, respectively, are: n(N₂) = $\frac{75.5 g}{28.02 g mol⁻¹}$= 2.69 mol n(O₂) = $\frac{23.2 g}{32.00 g mol⁻¹}$ = 0.725 mol n(Ar) = $\frac{1.3 g}{39.95 g mol⁻¹}$ = 0.033 mol The total is 3.45 mol. The mole fractions are obtained by dividing each of the above amounts by 3.45 mol and the partial pressures are then obtained by multiplying the mole fraction by the total pressure (1.20 atm): | | N₂ | O₂ | Ar | |:----|:---|:---|:---| | Mole fraction: | 0.780 | 0.210 | 0.0096 | | Partial pressure/atm: | 0.936 | 0.252 | 0.012 | ## Checklist of Concepts 1. The physical state of a sample of a substance, is its form (solid, liquid, or gas) under the current conditions of pressure, volume, and temperature. 2. Mechanical equilibrium is the condition of equality of pressure on either side of a shared movable wall. 3. An equation of state is an equation that interrelates the variables that define the state of a substance. 4. 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. 5. An isotherm is a line in a graph that corresponds to a single temperature. 6. An isobar is a line in a graph that corresponds to a single pressure. 7. An isochore is a line in a graph that corresponds to a single volume. 8. A perfect gas is a gas that obeys the perfect gas law under all conditions. 9. Dalton's law states that the pressure exerted by a mixture of (perfect) gases is the sum of the pressures that each one would exert if it occupied the container alone. ## Checklist of Equations | Property | Equation | Comment | Equation Number | |:---|:---|:---|:---| | Relation between temperature scales | *T*/K = θ/°C + 273.15 | 273.15 is exact | 1A.1 | | Perfect gas law | *PV* = *nRT* | Valid for real gases in the limit *p* → 0 | 1A.4 | | Partial pressure | *P₁* = *x₁*P | Definition; valid for all gases | 1A.6 | | Mole fraction | *x₁* = *n₁*/n | Definition | 1A.7 | | | *n* = *n₁* + *n₂* +... | | |