CHAPTER 1 THE KINETIC THEORY OF GASES-PART? Wen-Bin Jian ( 簡紋濱 ) Department of Electrophysics National Chiao Tung University
1. Molecular Model of an Ideal Gas. Molar Specific Heat of an Ideal Gas. Adiabatic Processes for an Ideal Gas 4. The Equipartition of Energy 5. Distribution of Molecular Speed OUTLINE
Ideal gas model: 1. The number of molecules in the gas is large and the average separation among molecules is large compared with their size.. Newton s law of motion is strictly obeyed for each molecule, but molecules move randomly.. The molecules interact only by short-range force during elastic collisions. 4. The molecules make elastic collisions with the walls. 5. The gas under consideration is a pure substance; that is, all molecules are identical. N molecules in the cube of length d each molecule has a mass of m and velocity use impulse to estimate the force on the walls, assume an interval between successive collisions on the wall 1. MOLECULAR MODEL OF AN IDEAL GAS
Ideal gas model:, 1,, 1. MOLECULAR MODEL OF AN IDEAL GAS
Ideal gas model: compare with We will learn the concept later from the equipartition theory. one degree of freedom: three degree of freedom: The internal energy of an ideal gas: The root-mean-square speed of a molecule: 1. MOLECULAR MODEL OF AN IDEAL GAS
Example: Please calculate the root-mean-square speed of a hydrogen molecule at room temperature., 0.00, 8.14, 00 190 m/s Example: A tank of volume 0.00 m contains.00 mole of helium gas at 0.0 o C. Assuming the helium behaves like an ideal gas, (a) find the total internal energy of the gas. (b) What is the root-mean-square speed of the helium atoms?.0, 0 9.15.00 8.14 9.15 710 8.14 9.15 0.004 150 / 1. MOLECULAR MODEL OF AN IDEAL GAS
Equations for an ideal gas: Molar specific heat at constant volume: 0 0, / Molar specific heat at constant pressure: 0 0, 5. MOLAR SPECIFIC HEAT OF AN IDEAL GAS
The ratio of the two molar specific heats is defined: The specific heats of monatomic, diatomic, and triatomic gas: (J/mol K) (J/mol K) / He 0.8 1.5 8. 1.67 5/ H 8.8 0.4 8. 1.41 7/5 CO 7 8.5 8.5 1.1 9/7 Example: A cylinder contains.00 mol of helium gas at a temperature of 00 K. (a) If the gas is heated at constant volume, how much energy must be transferred by heat to the gas after it is heated to 500 K? (b) How much energy must be transferred by heat to the gas at constant pressure?.00, 500 00 00, /, 5/.00 / 00 7480 8.14.00 5/ 00 1500. MOLAR SPECIFIC HEAT OF AN IDEAL GAS
Adiabatic process: what is the P-V dependence? 0 00 1 0 ln ln ln ln ln ln adiabatic isothermal 00. ADIABATIC PROCESSES FOR AN IDEAL GAS
Example: The fuel-air mixture in the cylinder of a diesel engine at 0.0 o C is compressed from an initial pressure of 1.00 atm and volume of 800 cm to a volume of 60.0 cm. Assuming that the mixture behave as an ideal gas with = 1.4 and that the compression is adiabatic, find the final pressure and temperature of the mixture. 1, 800, 9.15, 60 In an adiabatic process, the P-V follows. 1.00 800. 60.0. 7.6 9.15 800. 60.0. 86. ADIABATIC PROCESSES FOR AN IDEAL GAS
The theory of the equipartition of energy: In the equilibrium condition, each degree of freedom contributes an average energy of / per molecule. Monatomic gas molecule: three degrees of freedom of translational motion Diatomic gas molecule: three degrees of freedom for translation motion two degrees of freedom for rotational motion two degrees of freedom for vibrational motion Molar specific heat of the diatomic gas system:, for molecules only in translational motion, for molecules in translational and rotational motion, for molecules in translational, rotational, and vibrational motion 4. THE EQUIPARTITION OF ENERGY
molar specific heat at constant pressure of normal hydrogen molecules U.S. Department of Commerce National Bureau of Standards, V41, P79 (1948). The temperature dependence implies the energy quantization for rotational and vibrational motions. It points out the difference between classical and quantum statistics. 9/ 8.95 / 7/ 6.96 / 00 K 5/ 4.97 / 4. THE EQUIPARTITION OF ENERGY
Probability and statistics: distribution functions (the probability of the occurrence times) occurrence times of the i th event:, distribution function: /, 1 If the value of the i th event is the average value is the average of the square of the value is the root-mean-square of the value is the standard deviation is Continuous distribution:, 1 1 the value is also a function of x, the average of the value is,, 5. DISTRIBUTION OF MOLECULAR SPEED
The occurrence probability of a molecule with kinetic energy follows the Maxwell-Boltzmann distribution: exp In a two dimensional system, the number of molecules is proportional to the ring area and the energy distribution function:, / In a three dimensional system, the number of molecules is 4 exp / Normalize it by: exp / 4 5. DISTRIBUTION OF MOLECULAR SPEED
Mathematical methods: exp exp,, exp exp Let s calculate, put the one-dimensional integration into a two-dimensional space, and use the polar coordinate 1 exp 5. DISTRIBUTION OF MOLECULAR SPEED 1 4 exp exp 1 4 exp exp 1 4 exp 4 1 4 exp
Number of molecules as a function of speed: exp exp 1 exp / & exp / k=? 1 4 1 4 8 4 / 4 / exp / 5. DISTRIBUTION OF MOLECULAR SPEED
The average speed of the gas molecules: / 4 exp / 1 exp exp 4 / exp 1 1 exp / 4 / / exp / exp 1 exp / / / 8 1 5. DISTRIBUTION OF MOLECULAR SPEED
The average kinetic energy of the gas molecules: / 4 exp / 1 5. DISTRIBUTION OF MOLECULAR SPEED / exp / exp, exp 1 4 exp 1 exp 1 exp / / 8 exp / / 8 / equipartition theory
The distribution function: 4 / exp / 0 00 K 1 mol N molecules 8 900 K 5. DISTRIBUTION OF MOLECULAR SPEED
Example: Nine particles have speeds of 5.0, 8.0, 1.0, 1.0, 1.0, 14.0, 14.0, 17.0, and 0.0 m/s. (a) Find the average speed. (b) What is the rms speed? (c) What is the most probable speed of the particles? 5.0 8.0 1.0 14.0 17.0 0.0 9 1.7 / 5.0 8.0 1.0 14.0 17.0 0.0 1. / 9 Distribution function 4 1 / 1 0 0 5 10 15 0 5 5. DISTRIBUTION OF MOLECULAR SPEED