161ASpring2010-Lecture5

161ASpring2010-Lecture5 - Ideal Gas Law The combination of...

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1 Ideal Gas Law The combination of these three laws gives the ideal gas law which is a special form of an equation of state , i.e., an equation relating the variables that characterize a gas (pressure, volume, temperature, density, ….). The ideal gas law is applicable to low-density gases. constant (fixed mass of gas) B pV T pV nRT pV Nk T p RT ρ = = = =
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2 Statistical Mechanics and Thermodynmaics In thermodynamics a small number of macroscopic variables completely describes the state of the system (N,P,T,V,…) In statistical mechanics, the dynamical state of the system is described by stating the position and momenta of every particle in the system Since there are so many particles (≈10 23) these can then be treated statistically.
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3 Statistical Mechanics & Kinetic Theory of Gases The dynamical state of the system is compleletely described by 6N variable where N is the number of particles. 3N variables for positions and 3N variables for momenta. The energy of the system is described by the Hamiltonian The dynamical state of the system then evolves according to Newton’s laws of motion.
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4 Kinetic theory of an Ideal Gas 1. Consider a large number N indistinguishable particles 2. With mass m that move with random velocities (each direction is equally probable and a distribution of speeds from zero to infinity) 3. Confined to a cubic box of length l 4. The particles interact elastically with the box walls (kinetic energy is conserved in the interaction) 5. They do not interact with eachother hence they occupy no volume
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5 Pressure For a single collision: (the x-component changes sign) A F p = v m v x t ) mv ( t p F x x x = = x x mv p 2 = t mv F x x = 2 The force exerted by the particles on the walls of the container is the time-averaged momentum transfer due to collisions between the particles and the walls: If the time between such collisions = t, then the average force on the wall due to this particle is: t F x <F x > average" time " means <
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6 Pressure and Kinetic Energy Pressure and Kinetic Energy of an Ideal Gas of an Ideal Gas Now consider collisions of gas molecules in a closed volume. x
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161ASpring2010-Lecture5 - Ideal Gas Law The combination of...

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