LECTURE 9 IDEAL GAS & KINETIC THEORTY Lecture Instructor: Kazumi Tolich Lecture 9 2 ¨ Reading chapter 13-3 to 13-4. ¤ Ideal gasses ¤ Kinetic theory Ideal gas 3 ¨ In an ideal gas, intermolecular interactions are negligible. ¤ The molecules obey Newton’s laws of motion, and move about in a random manner. ¤ The molecules collide elastically with each other or container walls, and have no other interactions. ¨ The behavior of real gasses is well approximated by an ideal gas. ¨ An equation of state for an ideal gas relating the pressure P, volume V, number of molecules N and the temperature T (in Kelvin) is given by PV = NkT where k = 1.38 × 10-23 J/K is called the Boltzmann constant. Ideal gas: 2 4 ¨ A more natural measure of the number of molecules is the mole. ¨ A mole of substance is the amount that contains Avogadro’s number, 23 NA=6.022 × 10 , of molecules. ¨ Atomic or molecular mass of a substance is the mass of one mole of that substance. ¨ The equation of state for an ideal gas in terms of the number of moles n is given by PV = nRT where R = 8.31 J/(molK) is the universal gas constant. Clicker question 1 & 2 5 Example: 1 6 ¨ In the morning, when the temperature is Ti = 288 K, a bicyclist finds that the absolute pressure in his tires is Pi = 505 kPa. That afternoon, he finds that the pressure in the tires has increased to Pf = 552 kPa. Ignoring the expansion of the tires, find the afternoon temperature. Isotherms 7 ¨ If the temperature and the number of molecules in a gas are held constant, an ideal gas follows Boyle’s law: PV = nRT PiVi = PfVf constant P = V ¨ These curves are called isotherms. Demo: 1 8 ¨ Boyle’s law ¤ Demonstration of Boyle’s law n The higher the volume, the lower the pressure n The lower the volume, the higher the pressure constant P = V Example: 2 9 ¨ A cylindrical flask is fitted with an airtight piston that is free to slide up and down. A mass rests on top of the piston. Contained within the flask is an ideal gas at a constant temperature of T = 313 K. Initially the pressure applied by the piston and the mass is Pi = 137 kPa, and the height of the piston above the base of the flask is hi = 23.4 cm. When additional mass is added to the piston, the height of the piston decreases to hf = 20.0 cm. Find the new pressure applied by the piston, Pf. Constant pressure 10 ¨ If the pressure and the number of molecules in a gas are held constant, an ideal gas follows Charles’s law: V nR = T P V V i = f Ti Tf V = (constant)T Example: 3 11 ¨ A cylindrical flask is fitted with an airtight piston that is free to slide up and down. A mass rests on top of the piston. The initial temperature of the system is Ti = 313 K, and the pressure of the gas is held constant at P = 137 kPa. The temperature is now increased until the height of the piston rises from hi = 23.4 cm to hf = 26.0 cm. Find the final temperature of the gas, Tf. Demo: 2 12 ¨ Molecular motion demonstrator ¤ Demonstration of random molecular motion Pressure in the kinetic theory of gasses 13 ¨ The kinetic theory of gasses relates microscopic quantities (position, velocity) to macroscopic ones (pressure, temperature). ¨ Pressure is the result of collisions between the gas molecules and the walls of the container. ¨ The pressure of a gas P is directly proportional to the average kinetic energy of its molecules Kav. 2! N $ 2! N $! 1 2 $ P = # &Kav = # &# mv & 3"V % 3"V %" 2 %av N (N >> 1) is the number of molecules, V is the volume of the gas, m is the mass of a molecule, and v is the speed of a molecule. Internal energy of an ideal gas 14 ¨ The average kinetic energy of a gas molecule is directly proportional to the Kelvin temperature T. ¤ The hotter the temperature, the faster on average the molecules are moving. ! 1 2 $ 3 # mv & = Kav = kT " 2 %av 2 ¨ The internal energy of a substance is the sum of its potential and kinetic energies. ¨ For an ideal gas of N point-like molecules, the internal energy is from their translational kinetic energy. 3 3 U = NkT = nRT 2 2 Speed distribution of molecules 15 ¨ The molecules in a gas have a range of speeds. ¨ The overall distribution of speeds for molecules (the Maxwell-Boltzmann distribution) remain constant for a given gas. ¨ The Maxwell distribution indicates which speeds are most likely to occur in a given gas. ¨ For gas molecules, the rms speed, vrms, is given by 3kT 3RT v = v2 = = rms ( )av m M Clicker question: 3 16 Example: 4 17 ¨ One mole of oxygen molecule has a mass of 32.0 g. For oxygen in air at a room temperature of 20.0 °C, calculate the followings. a) The average translational kinetic energy b) The rms speed Example: 5 18 ¨ The rms speed of a sample of gas is increased by one percent. What is the percent change in temperature of the gas? .