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Microscopic model of ideal gas 1. The ideal gas law Microscopic description of an ideal gas includes the following assumptions: (1) Ideal gas consists of N identical molecules ; (2) The motions of molecules obey Newton’s law ; (3) The average distance between molecules are much larger than molecular size ; (4) Collisions between molecules or between molecules and the walls of container are elastic . Equilibrium state is a state in which macroscopic variables have definite values . It is a dynamic equilibrium . · To describe equilibrium state , P we introduce state variables ( P,V ) ( such as : P , V , T , m … ) . · We can express an equilibrium state in a PV diagram . o The equation ( ideal gas law ) that describes the state of an ideal gas can be written as : PV NkT kB=1.38×10–23 JK-1 is called Boltzmann’s constant V 2. Pressure P 289 - 292 Imagine the motion of molecules in a closed container , as shown . The constant , rapid drumbeat of molecules exerts a steady average force on the walls . Let’s calculate how that force is related to quantities which specify the motion of the molecules , namely , mass and velocity . Consider a single collision . A molecule of mass m approaches the wall with velocity v , and bounces off like a rubber ball . The change in momentum of the molecule is : P 2mvx According to Newton’s second law the force on the molecule is equal to the rate of change of y momentum . P 2mvx t t mv ' By Newton’s third law , the force the molecule exerts on the wall is : x mv 2mvx f z t Now suppose for a moment that the gas is extremely rarefied , so a typical molecule travels back and forth across the box many times before colliding with other molecule . y A molecule travels back and forth across the box once every t . mv ' t 2Lx / vx 2 x mv 2mvx mvx so f z t Lx summing over all the molecules in the box , we obtain the total force on the wall . N mvxi2 F i 1 Lx 2 xi N mv 1 F Lx i 1 Lx N mv i 1 2 xi Since the motion of the molecules is random , we have : 1 2 2 2 2 vxi v yi vzi vi 3 2 1 so F 3 Lx N 1 2 2 K mvi 3 Lx i 1 2 The pressure on the wall is the average force divided by the area . F 2 K 2 K 2 NK P A 3 Lx Ly Lz 3 V 3 V 3. Temperature and internal energy The pressure can be written as : P 297 - 298 2 NK P 3 V The ideal gas law is given by : PV NkT 3 So , we have K kT 2 It is average kinetic energy for monatomic gases . The total energy ( called internal energy ) of a monatomic gas is : 3 U NkT 2 For ideal monatomic gases : 3 U NkT 2 For ideal diatomic gases : 5 U NkT 2 For ideal polyatomic gases : 6 U NkT 2 For ideal gases : U qNkT