10 TEMPERATURE, THERMAL EXPANSION, IDEAL GAS LAW, AND KINETIC THEORY OF GASES.
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1 10 TEMPERATURE, THERMAL EXPANSION, IDEAL GAS LAW, AND KINETIC THEORY OF GASES. Key words: Atoms, Molecules, Atomic Theory of Matter, Molecular Mass, Solids, Liquids, and Gases, Thermodynamics, State Variables, Temperature, Thermometers, Temperature Scales, Celsius and Fahrenheit Temperature Scales, Absolute or Kelvin Temperature Scale, Thermal Expansion, Linear Expansion, Coefficient of Linear Expansion, Bridge Expansion, Volume Expansion, Coefficient of Volume Expansion, Equation of State, Boyle s Law, Charles s Law, Charles s Law, Mole, Universal Gas Constant, Ideal Gas Law or the Equation of State of an Ideal Gas, Standard Temperature and Pressure, Compressing Gas in an Automobile Engine, Avogadro s Number, Boltzmann s constant, Kinetic Theory, Molecular Interpretation of Temperature, Root-Mean-Square Speed, Maxwell Boltzmann Distribution. Now we make deep insight in the structure of matter and then based on this insight we will try to develop approach that allows describe important properties of matter. Atomic Theory of Matter. Microscopic and Macroscopic Description of Matter. Thermodynamics. The atomic theory of matter postulates that that all matter is made up of tiny entities called Atoms, which a typically 10^(-11)- m in diameter. Some substances are composed of the groups of atoms that are called Molecules. The atoms and molecules are characterized also by atomic and Molecular Masses. They are extremely small and measured in atomic mass units u. 1 u = ^(-27) kg. The distinction between Solids, Liquids, and Gases can be attributed to the strength of the attractive forces between the atoms or molecules and to their average speed. In gases, for example, the interatomic forces so weak and atomic speeds so high, that atoms or molecules do not even stay close together. They move freely and fill any container. When two molecules collide, they fly off in new direction. We can describe the state (or condition) of a particular system such a gas in a container from either a microscopic or macroscopic point of view. A
2 microscopic description would involve details of the motion of all atoms or molecules making up the system. It could be very complicated for macroscopic systems composed of huge numbers of atoms or molecules. A macroscopic description is given in terms of quantities that are detectable directly by our senses, such as volumes, mass, pressure and temperature. The description of processes in terms of these macroscopic quantities is the field of branch of Physics that we start to study: Thermodynamics. In contrast to Mechanics, in which we characterize objects or system of objects by variables like coordinates, velocities, forces and their changes in time, in thermodynamics we use other variables. The number of variables depends on type of system. To describe the state of a pure gas in a container, for example, we need only three variables, which could be the volume V, the pressure P, and the temperature T. Quantities such as these that can be used to describe the state of a system are called State Variables. In mechanics, we used space coordinates to locate the object position in space and we used graphical tools to study how mechanical variables are changing in time. In Thermodynamics, we will use other tools: diagrams showing relation between state variables. For example, to describe gases behavior we will use P V, P T, and V T diagrams. We will pay most our attention to gases. On the one hand, they are simplest thermodynamic systems that could be consistently studied in our course. On the other hand, gases are used as working substances in the heat engines. Our civilization is based on the heat engines. Important part of thermodynamics dedicated to study of heat engines, and we will try also learn basic concepts of thermodynamic description of them. First of all, we consider one of the basic concepts of thermodynamics -- the temperature Temperature, Thermometers, and Temperature Scales. Temperature is a measure of how hot or cold a body is. Instrument designed to measure temperature are called Thermometers. There are many kinds of thermometers, but their operation always depends on some property of matter that changes with temperature. Most common thermometers rely on the expansion of a material with an increase in temperature. The first idea for a thermometer, by Galileo, made use of the expansion of a gas. Very precise thermometers make use of electrical properties, such as resistance thermometers, thermocouples, and thermistors, which may have a digital readout.
3 In order to measure temperature quantitatively, some sort of numerical scale must be defined. Celsius and Fahrenheit Temperature Scales are based on the freezing (0ºC = 32ºF) and boiling (100ºC = 212ºF) temperatures of water, related by the formulas TF = (9/5) TC + 32º (10-1) TC = (5/9) (TF -- 32º) (10-2) The Celsius scale sometimes is called the Centigrade scale. In science, the most important scale is the Absolute or Kelvin Temperature Scale. On this scale the temperature is specified as degrees Kelvin or preferable, simply as kelvins (K) without the degree sign. The intervals are the same as the Celsius scale. The Kelvin scale has its zero at the extrapolated zero-pressure temperature for a gas thermometer, which is ºC. Thus 0 K = ºC (10-3) TK = TC (10-4) The 1 K is the unit to measure temperature in the SI system. Thermal Expansion. Most substances expand when heated and contract when cooled. The change in length of any linear dimension L0 of a solid object with a temperature change is called Linear Expansion. Experiments indicate that the change in length ΔL of almost of all solids, to a good approximation, directly proportional to the change in temperature ΔT, as long as ΔT is not too large. The change in length is different for different materials. We can write the proportionality between ΔL and ΔT as an equation: ΔL = α L0 ΔT (10-5) Where α is the Coefficient of Linear Expansion for the particular material and has units of (Cº)^(--1). For example, coefficient of expansion of Aluminum at 20ºC is ^(--5) (Cº)^(--1) or ^(--5) (K)^(--1). Thermal expansion plays an important role in numerous engineering applications. For example, thermal expansion joints must be included in bridges and some other structures to compensate for changes in dimensions
4 with temperature variations. Without these joints, the surfaces would buckle due to thermal expansion on very hot days or crack due to contraction on very cold days. EXAMPLE Bridge Expansion. The Verrazano Narrow Bridge in New York City is one of the world's longest suspension bridges. The allowance for its expansion in the temperature range of ΔT = 120 K is 1.87 m. (a) What is the length of this bridge? (b) How much expansion must be allowed for the central span of the Golden Gate Bridge near San Francisco (central span length is 1280m for the same ΔT? Assume that α = ^(-5) K^(-1) for both of these bridges. ΔT = 120 K ΔL1 = 1.87 m L02 = 1280m α = ^(-5) K^(-1) (a) L01 --? (b) ΔL2 --? (a) We can write the equation (10-5) in this situation as follows: ΔL1 = α L01 ΔT L01 = ΔL1 /(α ΔT) = 1300 m (b) For Golden Gate Bridge the equation (10-5) can be written as follows: ΔL2 = α L02 ΔT ΔL2 = [1.2 10^(-5) K^(-1)] (1280m) (120 K) = 1.84 m The change in volume ΔV in the volume V0 of any solid or liquid material with a temperature change ΔT is called the Volume Expansion. It is described by a relation similar to equation (10-5), namely, ΔV = β V0 ΔT (10-6) Where β is called the Coefficient of Volume Expansion for the particular material and has units of K^(--1). For example, coefficient of expansion of gasoline at 20ºC is ^(--4) (K)^(--1) or ^(--4) (Cº)^(--1). Values of β for liquids much larger than those for solids. For most of solids β 3 α. Water is unusual substance because, unlike most materials whose
5 volume increases with temperature, its volume actually decreases as the temperature increases in the range of temperature from 0^C to 4^C Ideal Gas Law. Equation (10-6) describing the expansion of solids and liquids is not very useful for describing the expansion of a gas partly because gases generally expand to fill whatever container they are in. The volume of gas depends very much on the pressure as well as on the temperature. It is therefore valuable to determine a relation between the volume V, the pressure P, the temperature T, and the mass m of a gas. Such relation is called an Equation of State. (By the word state, we mean the physical condition of the system.) We start with experimentally determined laws for a given quantity of gas. They relates two from these four variables: P and V; V and T; P and T. They are called correspondingly: Boyle s Law V 1/P PV = const (10-7) Charles s Law V T (10-8) Charles s Law P T (10-9) Experiments show also that at constant temperature and pressure V m. Combining all these relationships, we can write P V m T (10-10) This proportion can be made into an equation by inserting a constant of proportionality specific for each gas. But, it is very interesting that the constant of proportionality turns out to be the same for all gases, if instead of the mass m, we use the number of moles. One Mole (1 mol) is the amount of substance that contains as many elementary entities (atoms or molecules) as there are in 12.0 grams of carbon 12 (whose atomic mass is exactly 12 u). The mole is the unit of the amount of substance in the SI system. We can transform now proportion (10-10) into equation:
6 PV = n R T (10-11) Where n represents the number of moles and R is called the Universal Gas Constant. R is the same for all gases. In SI system R = J / (mol K). The equation (10-11) is called Ideal Gas Law or the Equation of State of an Ideal Gas. Generally speaking, the relationship among P, V, T, and n for any substance is called equation of state. In the case of ideal gas it is very simple and can be written as the equation (10-11) for all ideal gases. We use the term ideal because real gases do not follow equation (10-11) precisely. However, at pressures less than an atmosphere or so, and when T is not close to the liquefaction temperature of a gas, equation (10-11) is quite accurate and useful for real gases. Always remember, when using the ideal gas law, that temperature T is absolute temperature and must be given in kelvins (K), and that pressure is absolute pressure. The ideal gas law is an extremely useful tool to solve problems. EXAMPLE Volume of one mole of an ideal gas at STP. The condition called STP (Standard Temperature and Pressure) for a gas is defined to be a temperature T = 0 C = K and a pressure P = 1 atm = ^5 Pa. m = 1 mol T = 0 C = K P = 1 atm = ^5 Pa R = J / (mol K) V --? From (10-11): PV = n R T V = n R T/ V V = m³ = 22.4 L Thus the volume of one mole of any ideal gas is 22.4 liters. This is almost exactly the volume of 3 basketballs. Many problems involve a change in the pressure, temperature and volume of a fixed amount of gas. In this case, for any two state of a gas 1 and 2 n1 = n2. From equation (10-11) for both of this state we can write: Finally we will get P1 V1 / T1 = n1 R = n2 R = P2 V2 / T2 (10-12)
7 P1 V1 / T1 = P2 V2 / T2 (10-13) Therefore, in the case n =const, we have conservation of the following combination of state variables of an ideal gas P V / T = const (10-14) As a result we can use equation (10-13) to predict unknown values of the state variables of the gas in problems. EXAMPLE Compressing Gas in an Automobile Engine. In an automobile engine, a mixture of air and gasoline is compressed in the cylinder before being ignited. A typical engine has a compression ratio of 9.00 to 1; this means that the gas in the cylinders is compressed to 1 / (9.00) of its original volume. The initial pressure is 1 atm, and initial temperature is 27 C. If the pressure after compression is 21.7 atm, find the temperature of the compressed gas. V1 /V2 = 9 P1 = 1.00 atm P2 = 21.7 atm T1 = 27 C T2 --? From P1 V1 / T1 = P2 V2 / T2 T2 = P2 V2 / (P1 V1) = [(21.7 atm) V2] / [(1.00 atm) V1] T2 = 723 K = 450 C Ideal Gas Law in terms of Molecules: Avogadro s Number. The Italian scientist Avogadro stated that equal volumes of gas at the same pressure and temperature contain equal numbers of molecules. This is consistent with the experimental observation that universal gas constant is the same for all ideal gases. The number of molecules in 1 mol of each ideal gas must be the same and this number was called after Avogadro. Thus,
8 Avogadro s number, NA = ²³ molecules/( per mole), is the number of atoms or molecules in 1 mol of any pure substance. The Ideal Gas Law can be written now in terms of the number of molecules N in the gas as P V = N k T (10-15) where k = R / NA = ²³ J/K is the Boltzmann s constant. Now we try to connect macroscopical description of an ideal gas and microscopical one. The equation (10-15) shows that we can think of k as a gas constant on a per molecule basis instead of the usual per mole basis with the gas constant R Kinetic Theory of an Ideal Gas and the Molecular Interpretation of Temperature. The goal of any molecular theory of matter is to understand the macroscopic properties of matter in terms of its atomic or molecular structure and behavior. Such theories are of tremendous practical importance; once we have understanding, we can design materials to have specific desired properties. The assumptions of a molecular model of an ideal gas as follows: 1. A gas is composed of a very large number N of identical molecules, each with mass m. 2. The molecules behave as point particles. 3. The molecules are in constant random motion. Their motion obeys the Newton s law of motion. The collisions between molecules and between molecules and walls of the container are perfectly elastic, conserving the molecules momentum and energy. 4. During collisions, the molecules exert force on the walls of the container; these forces create the pressure that the gas exerts. It could be shown that according to kinetic theory of ideal gas, the average translational kinetic energy Kav of molecules in random motion in an ideal gas is directly proportional to the absolute temperature T of the gas: Kav = ½ m (vav)² = (3/2) kt (10-16) Where m is the mass of an atom or molecule, vav is the average speed of the atom or molecule. This result shows that the average translational kinetic energy per molecule directly proportional to the absolute temperature of the gas. It is very interesting that, in ideal gas, the average translational kinetic energy per molecule depends only on the temperature, not on pressure,
9 volume, or kind of molecule involved. The equation (10-16) holds not only for gases, but also applies reasonably accurately to liquids and solids. Therefore we can say that the temperature of an object is a measure of the average kinetic energy of atoms or molecules by which this macroscopic object is made. From (10-16) we can obtain an expression for the square root of (vav)², called the Root-Mean-Square Speed vrms of molecules in an ideal gas. vrms = (vav)² = 3kT / m = 3RT / (m NA) = 3RT / M (10-17) where M = m NA is the mass of one mole of a substance or the Molar mass. To compute the rms speed, we square each molecular speed, add, divide by the number of molecules, and take the square root. That is, vrms is the root of the mean of the squares. Actually, at any moment, there is a wide distribution of molecular speeds within a gas. This distribution is called Maxwell Boltzmann Distribution after Scottish physicist Maxwell and Austrian physicist Boltzmann that made a great contribution to the kinetic theory of gases. The distribution exhibits a single peak; this is the most probable value for a molecule s speed. The distribution curve is not symmetrical about the most probable speed because the lowest speed must be zero while there is no classical limit to the highest speed. The mean speed is therefore larger than the most probable speed. The root-mean-square speed, which involves the mean of the squares, is still larger. At higher temperatures, the range of typical speeds is greater, and the distribution is broader. The number of molecules that have speeds greater than some given speed increases as the temperature increases. The Maxwell -- Boltzmann distribution shows why chemical reaction rates increase dramatically with temperature. It is the most energetic molecules, of which there are more at high temperatures that are largely responsible for sustaining chemical reactions. This also explains many other phenomena, such, for example, the increase in the rates of certain nuclear reactions with rising temperature. EXAMPLE Molecular Kinetic Energy and molecular speed. Calculate: (a) the average kinetic energy of a molecule in air at room temperature; (b) the vrms speed of nitrogen molecule with this energy. T = 20 C = 293 K mn2 = 2 (14 u) = 28 u u = ^(-27) kg
10 M = m NA =28 u ²³ /mol= ^(-2) kg/mol k = ^(-23) J/K R = J/(mol K) (a) Kav --? (b) vrms --? (a) The molecule of nitrogen gas is composed of two nitrogen atoms, so its mass mn2 two times greater than the mass of one nitrogen atom mn. We can take the atomic mass of nitrogen atom from the Periodic Table: mn = 14 u. Then mn2 = 2 mn. The average kinetic energy Kav is given by equation (10-16), in which T is the absolute temperature, so does not forget to express temperature in kelvins. Finally, we will get: Kav = ½ mn2 (vav) ² = (3/2) kt = (3/2) [ ^(-23) J/K] (293 K) = ^(-21) J (b) To find vrms, we will use equation (10-17): vrms = (vav)² = 3kT / mn2 = 3RT / (m NA) = 3RT / M If we will use microscopic approach then vrms = 3kT / mn2 = 3 [ ^(-23) J/K] (293 K)/{28 [ ^(-27) kg]} vrms = 511 m/s In macroscopic approach, we need to know the molar mass M = m NA =28 u ²³ /mol= ^(-2) kg/mol = 28 g/ mol. If we will use now the macroscopic approach to find vrms we will get the same result, as we have got using microscopical approach. vrms = 3RT / M = 3 [8.314 J/(mol K)] (293 K) / [2.8 10^(-2) kg/mol] vrms = 511 m/s Not surprisingly, this thermal speed is of the same order of magnitude as the sound speed (340 m/s) in air at room temperature. At the microscopic level, the speed of the individual molecules sets an approximate upper limit on the maximum rate at which information can be transmitted by disturbances that is, sound waves propagating through the gas.
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