Thermal Properties of Matter PDF

THERMAL PROPERTIES OF MATTER PHY 103: Thermodynamics 1 PROPERTIES OF MATTER MACROSCOPIC MICROSCOPIC (large-scale) (small-scale) Pressure Speed Volume Kinetic energy Temperature...

THERMAL PROPERTIES OF MATTER PHY 103: Thermodynamics 1 PROPERTIES OF MATTER MACROSCOPIC MICROSCOPIC (large-scale) (small-scale) Pressure Speed Volume Kinetic energy Temperature Momentum Mass Molar mass 2 EQUATION OF STATE ❖ The conditions in which a particular material exists are described by physical quantities such as pressure (p), volume (V), temperature (T), and amount of substance (m or n). ❖ These variables describe the state of the material and are called state variables. ❖The relationship among these variables (p, V, T, and m) is simple enough that we can express it as an equation called the equation of state. 3 IDEAL-GAS EQUATION A simple equation of state. (1) 4 In terms of mass (mtotal) of the gas, using mtotal = nM, the ideal-gas equation can be expressed as: (2) Hence, the density (ρ = mtotal/V) of the gas is given by: (3) 5 For a constant mass (or constant number of moles), Thus for two different systems with the same amount of gas, (4) 6 SAMPLE PROBLEM 1: What is the volume of a container that holds exactly 1 mole of an ideal gas at standard temperature and pressure (STP), defined as T = 0oC = 273.15 K and p = 1 atm = 1.013 x 105 Pa? Solution: 7  SAMPLE PROBLEM 2: In an automobile engine, a mixture of air and vaporized gasoline is compressed in the cylinders before being ignited. A typical engine has a compression ratio of 9.00 to 1; that is, the gas in the cylinders is compressed to 1/9.00 of its original volume. The intake and exhaust valves are closed during the compression, so the quantity of gas is constant. What is the final temperature of the compressed gas if its initial temperature is 27oC and the initial and final pressures are 1.00 atm and 21.7 atm, respectively? 8 SAMPLE PROBLEM 3: An “empty” aluminum scuba tank contains 11.0 L of air at 21°C and 1 atm. When the tank is filled rapidly from a compressor, the air temperature is 42°C and the gauge pressure is 2.10 x 107 Pa. What mass of air was added? (Air is about 78% nitrogen, 21% oxygen, and 1% miscellaneous; its average molar mass is 28.8 g/mol = 28.8 x 10-3 kg/mol.) 9 SEATWORK 1: A 20.0-L tank contains 4.86 x 10-4 kg of helium at 18.0oC. The molar mass of helium is 4.00 g/mol. (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres? 10  IDEAL GAS ASSUMPTIONS 1. The total volume of the gas molecules is much less than the volume of the container they’re in. 2. There is no intermolecular interactions, i.e. no potential energy. 3. The collisions are perfectly elastic, which means that no energy is lost. 11  IDEAL GAS ASSUMPTIONS 4. The size of the molecules are very small in comparison to the average distance between particles and to the dimension of the container. 5. The container walls are perfectly rigid, infinitely massive with no tendency to move. 12 Ideal-gas Model vs more realistic model of gas 13  van der Waals Equation - developed by a Dutch physicist, Johannes Diderik van der Waals. (5) a and b are empirical constants and are different for different gases a – depends on the attractive intermolecular forces b – represents the volume of a mole of molecules 14 *When n/V is small (that is, when the gas is dilute), the average distance between molecules is large, the corrections in the van der Waals equation become insignificant, and the equation reduces to the ideal-gas equation  pV-Diagram of Ideal Gas (for constant amount of an ideal gas) Isotherm – constant-temperature curves Boyle’s Law: 15 pV-Diagram of Non-Ideal Gas Below Tc ❑ The isotherm develop a flat region. ❑ The material condenses to liquid as it is compressed. Critical temperature Above Tc ❑ There is no liquid-vapor phase transition. 16 MOLECULES Any specific chemical compound is made up of identical molecules. In gases the molecules move nearly independently. In liquids and solids, molecules are held together by intermolecular forces. These forces arise from the interactions among electrically charged particles that make up the molecules. Gravitational forces between molecules are negligible in comparison with electrical forces. 17 INTERMOLECULAR FORCES 18  INTERMOLECULAR FORCES  In solids, molecules vibrate about more or less fixed points.  In a liquid, the intermolecular distances are usually only slightly greater than in the solid phase of the same substance, but the molecules have much greater freedom of movement. In gases, molecules are usually widely separated and so have very small attractive forces. 19  MOLES AND AVOGADRO’S NUMBER One mole of any pure chemical element or compound contains a definite number of molecules, the same number for all elements and compounds. Official SI definition: One mole is the amount of substance that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12. The number of molecules in a mole is called Avogadro’s number, NA. Molar mass: (6) 20  EXAMPLE: The atomic mass of atomic hydrogen is MH = 01.008 g/mol and the atomic mass of oxygen is MO =16.0 g/mol. Find the mass of a single hydrogen atom and of a single oxygen molecule (O2). Solution: 21  KINETIC-MOLECULAR MODEL OF AN IDEAL GAS ASSUMPTIONS: 22  COLLISION WITH THE WALL During collisions the molecules exert forces on the walls of the container; this is the origin of pressure that the gas exerts. (7) m – mass of one gas molecule 23  KINETIC ENERGY Average translational kinetic energy of n moles of ideal gas: (8) To get the kinetic energy of one molecule, we divide Ktr by the total number of molecules, N: m – mass of one gas molecule 24 Also, the total number of molecules N is the number of moles n multiplied by Avogadro’s number NA so (9) Then, Hence, the average translational kinetic energy of one gas molecule is: k – Boltzmann constant (10) **This shows that the average translational kinetic 25 energy per molecule depends only on temperature. To obtain the average translational kinetic energy per mole, multiply equation 10 by Avogadro’s number (11) Using N = NAn and R = NAk, the ideal-gas equation can also be expressed as: (12) 26  MOLECULAR SPEED From equations 10 and 11, we can obtain the root-mean-square speed (or rms speed) of a gas molecule: (13) 27  SAMPLE PROBLEM 4 (a) What is the average translational kinetic energy of an ideal gas molecule at 27oC? (b) What is the total random translational kinetic energy of the molecules in 1 mole of this gas? (c) What is the root-mean-square speed of oxygen molecules at this temperature? 28 Sample Problem 5b REVIEW Equations of state Ideal-Gas Equation: pV = nRT van der Waals Equation: Molecular properties of matter mtotal – total mass n – number of moles M – molar mass NA – Avogadro’s number m – mass of one molecule Kinetic-molecular model of an ideal gas 29  COLLISIONS BETWEEN MOLECULES If gas molecules are really points, they never collide. But consider a more realistic model in which gas molecules are rigid spheres with radius r. How often do they collide with other molecules? How far do they travel, on average, between collisions? COLLISIONS BETWEEN MOLECULES Mean free path (λ) of a gas molecule: - average distance traveled between collisions (14) tmean – mean free time; average time between collisions 31  SAMPLE PROBLEM 5 (a) Estimate the mean free path of a molecule of air at 27oC and 1 atm. Model the molecules as spheres with radius r = 2.0 x 10-10 m. (b) Estimate the mean free time of an oxygen molecule with v = vrms at 27oC and 1 atm. 32 (a) Estimate the mean free path of a molecule of air at 27oC and 1 atm. Model the molecules as spheres with radius r = 2.0 x 10-10 m. Solution: Recall that Hence, 33 (b) Estimate the mean free time of an oxygen molecule with v = vrms at 27oC and 1 atm. Solution: From Sample Problem 4c, vrms = 484 m/s for oxygen molecule. 34  HEAT CAPACITIES Recall For solids, measurements are cp usually made at constant atmospheric Cp pressure. For gas, it is usually easier to keep the substance in a container with constant volume (cV ,CV) 35  HEAT CAPACITIES OF GASES When the temperature changes by a small amout dT, the corresponding change in kinetic energy is From the definition of molar heat capacity at constant volume, we also have 36  HEAT CAPACITIES OF GASES If Ktr represents the total molecular energy, as we have assumed, then dQ and dKtr must be equal. Hence (15) 37 Motions of a diatomic molecule When the temperature is increased in a diatomic or polyatomic gas, additional heat is needed to supply the increased rotational and vibrational energies. Thus polyatomic gases have larger molar heat capacities than monoatomic gases. 39 Principle of equipartition of energy - states that each velocity component (either linear or angular) has, on average, an associated kinetic energy per molecule of ½ kT. The number of velocity components needed to describe the motion of a molecule completely is called the number of degrees of freedom. If we assign five degrees of freedom to a diatomic molecule, the average total kinetic energy is 5/2 kT instead of 3/2 kT. The total kinetic energy of n moles is and the molar heat capacity is (16) 40 41 HEAT CAPACITIES OF SOLIDS The average kinetic energy of an atom is equal to its average potential energy. KEave = 3/2 kT (3 degrees of freedom) PEave = 3/2 kT Hence, for a crystal with N atoms or n moles, Etotal = 3NkT = 3nRT Thus, (17) 42 MOLECULAR SPEEDS 43 THE MAXWELL-BOLTZMANN DISTRIBUTION (18) In terms of translational kinetic energy of a molecule, (19) 45 46 PHASES OF MATTER Ordinary matter exists in the solid, liquid, and gas phases. A phase diagram shows conditions under which two phases can coexist in phase equilibrium. All three phases can coexist at the triple point. The vaporization curve ends at the critical point, above which the distinction between the liquid and gas phases disappears. 47  SHORT QUIZ _____ 1. A cylinder contains oxygen gas at a temperature of 7ºC and a pressure of 15 atm in a volume of 100 L. A fitted piston is lowered into the cylinder, decreasing the volume occupied by the gas to 80 L and raising the temperature to 40ºC. The gas pressure is now approximately (a) 17 atm (b) 21 atm (c) 108 atm (d) 595 atm (e) 3040 atm 49  SHORT QUIZ _____ 2. In a vacuum system, a container is pumped down to a pressure of 1.33  10–6 Pa at 20ºC. How many molecules of gas are there in 1 cm3 of this container? (a) 3.3 x 108 (b) 4.8  109 (c) 3.3  1014 (d) 7.9  1012 (e) 4.8  1012 50  SHORT QUIZ _____ 3. The air in a balloon occupies a volume of 0.10 m3 when at a temperature of 27ºC and a pressure of 1.2 atm. What is the balloon's volume at 7ºC and 1.0 atm? (The amount of gas remains constant.) (a) 0.022 m3 (b) 0.078 m3 (c) 0.089 m3 (d) 0.11 m3 (e) 0.13 m3 51  SHORT QUIZ ______4. When the temperature of an ideal gas is increased from 300 K to 400 K, the average kinetic energy of the gas molecules increases by a factor of (a) 1.15 (b) 1.33 (c) 1.54 (d) 1.78 (e) 2.47 52  SHORT QUIZ ______ 5. Two monoatomic gases, helium and neon, are mixed in the ratio of 2 to 1 and are in thermal equilibrium at temperature T (the molar mass of helium = 4.0 g/mol and the molar mass of neon = 20.2 g/mol). If the average kinetic energy of each helium atom is 6.3  10−21 J, calculate the temperature T. (a) 304 K (b) 456 K (c) 31 K (d) 31˚C (e) 101 K 53  SHORT QUIZ BONUS PROBLEM (5 points) How Many Atoms Are You? Estimate the number of atoms in the body of a 50-kg physics student. Note that the human body is mostly water, which has molar mass 18.0 g/mol, and that each water molecule contains three atoms. 54  SAMPLE PROBLEM 6 A flask with a volume of 1.50 L, provided with a stopcock, contains ethane gas at 300 K and atmospheric pressure (1 atm = 1.013 x 105 Pa). The molar mass of ethane is 30.1 g/mol. The system is warmed to a temperature of 380 K, with the stopcock open to the atmosphere. The stopcock is then closed, and the flask cooled to its original temperature. (a) What is the final pressure of the ethane in the flask? (b) How many grams of ethane remain in the flask? 55 Ans. 8.0 x 104 Pa; 1.45g  SAMPLE PROBLEM 7 A balloon whose volume is 750 m3 is to be filled with hydrogen at atmospheric pressure (1 atm = 1.013 x 105 Pa). The molecular mass of hydrogen is 2.02 g/mol. (a) If the hydrogen is stored in a cylinder with volume of 1.90 m3 at a gauge pressure of 1.20 x 106 Pa and temperature at 300 K, how many cylinders are required? Assume that the temperature of hydrogen remains constant. (b) Calculate the mass of the hydrogen in the balloon. Gauge pressure – pressure relative to atmospheric pressure. Absolute pressure – sum of gauge pressure and atmospheric pressure. pabs = pg + patm 56 Ans. 30.7 cylinders; 61.53 kg  SAMPLE PROBLEM 8 A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 1.00 atm at 20.0oC. The round part of the tank has a radius of 10.0 cm, and the gas is supporting a piston that can move up and down in the cylinder without friction. (a) What is the mass of the piston? (b) How tall is the column of the gas that is supporting the piston? 57 Ans. 325 kg; 1.38 m  SAMPLE PROBLEM 9 The speed of propagation of a sound wave in air at 27oC is about 350 m/s. Calculate for comparison, (a) vrms for nitrogen molecules and (b) the rms value of vx at this temperature. The molar mass of nitrogen (N2) is 28.0 g/mol. 58 Ans. 517 m/s; 298.6 m/s  SAMPLE PROBLEM 10 A deuteron is a nucleus of a hydrogen isotope and consists of one proton and one neutron. The plasma of deuterons in nuclear fusion reactor must be heated to about 300 million K. (a) What is the rms speed of the deuterons? Is this a significant fraction of the speed of light (c = 3.0 x 108 m/s)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10c? (Mass of proton = 1.673 x 10-27 kg and mass of neutron = 1.675 x 10-27 kg) 59 Ans. 1.93 x 106 m/s; 7.3 x 1010 K