Gas Behavior: Boltzmann Distribution and Kinetic Energy
Explore the concepts of Boltzmann distribution, average particle speed, average kinetic energy, and internal energy of a gas. Learn how statistical mechanics describes the distribution of particle speeds in an ideal gas and the relationship between kinetic energy, temperature, and pressure. Discover the roots of mean square velocity and how it varies with molar mass in gases.
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Physics 2 Feb 11, 2020 Get out xtra Calorimetry WS and p140 #13-21 for 2 Hmk Checks P3 Challenge A gas that has a volume of 28 liters, a temperature of 45 0C, and an unknown pressure has its volume increased to 34 liters and its temperature decreased to 35 0C. If I measure the pressure after the change to be 2.0 atm, what was the original pressure of the gas? Today s Objective: Internal Energy Assignment: Ch 3.2, p141, #22-32 Agenda including Show practice Gases revisit Homework reviews Boltzmann Distribution Average K.E. Internal Energy
Boltzmann Distribution Boltzmann distribution is a result of statistical mechanics that describes how the random particle speeds of an ideal gas are distributed. Note: Statistical mechanics is the application of statistics to the gas particle behavior. Unsymmetrical distribution: Always a finite probability at any high speed, but 0 probability at 0 speed. The ranking of increasing temperature for the three graphs are Blue<Red<Green. 1) the peak location increases and lowers 2) the speeds spread out.
Average Particle Speed Three types of average speed values for the particles of an ideal gas most probable speed (mode) left red peak average speed (mean) center blue c = root mean square speed corresponds to average kinetic energy (temp) right green peak Best indication of temperature is c because temperature is average kinetic energy IB assumes c represents all three. Unfortunately IB uses c for both vrms and the speed of light. Beware: This is NOT c = 3 x 108 m/s
Average Kinetic Energy 1 2 3 2 = 2 E m c The average K.E. is related to the average speed, c, where ma is the atomic mass of a single gas particle. Note on units: 1 u = 1.66 x 10-27 kg K a = E k T Derive from the ideal gas law and statistical mechanics: K B 1 3 = 2 = P dc PM dRT R N 23J = = 1.38 10 k K B And other basic relationships: m V A m M N N = d = = = M N m n A a A
Internal Energy of a Gas 3 2 If the average kinetic energy of one particle of gas is 3/2 kT, and you have a sample containing N particles, the internal energy of the sample is N times the average kinetic energy. From this, and recalling the definitions of kB and moles you can derive the expression for the internal energy of a gas, U. Problem solving is either of the plug and chug variety, or is algebra derivation of formulas types. So know these two fundamental relationships. (Only the first is in the IB packet.) Notice that both average EK and U only depend on T in K. = E k T K B N N R N = = n k B A A 3 2 3 2 = = U nRT PV
Root Mean Square Velocity Because kinetic energy depends on both mass and velocity, two gasses with the same kinetic energy can have different vrms. 3 rms v M RT = Where M is the molar mass of the gas. Root mean square velocity depends on the molar mass of the gas. Ex: Calculate the root mean square speed of a hydrogen molecule at STP. Ex: Calculate the most probable speed of a hydrogen molecule at STP.
Exit Slip - Assignment Exit Slip- What is the internal energy of a 1.60 m3 sample of helium gas at 115 kPa? What s Due? (Pending assignments to complete.) Ch 3.2, p140, #22-32 DOWNLOAD Engineering Physics Text B!!!! What s Next? (How to prepare for the next day) Start studying for the U6 Thermal Physics Test on next Thurs Feb 20
