momentum change per impact The average rate of change of momentum = Time interval between successive impacts 2m x 2l / x m x m x 2 / l P = l 2 P = l 3

Transcription

1 Kinetic Molecular Theory This explains the Ideal Gas Pressure olume and Temperature behavior It s based on following ideas:. Any ordinary sized or macroscopic sample of gas contains large number of molecules. These molecules are small compare to size of gas sample.. These molecules are in continuos and randomly directed motion. 4. They are independent of each other and interact with each other or walls of container only during the brief collisions. 5. These collisions, on the average, are perfectly elastic: i.e. no translational energy is loss because of collisions. 6. Newtonian mechanics, the equation f ma, can used to describe the average interaction of the molecules with walls of the container. Molecular Basis of Gas Pressure-One Molecule: Refer to Figure. Force exerted by a container wall to bounce molecule back when it collides with wall can be expressed by Newton's equation f ma where f force, m mass and a acceleration a d /dt f m ( d /dt) f d(m )/dt since mass of each molecule is constant. Thus, according to Newton's Law Applied force Rate of change of momentum Since velocity is a scalar quantity which shows speed and direction, it cans be resolved in three scalar components with magnitude x, y and z The magnitude of the momentum with which molecule approaches the wall is (i.e. x - direction) and magnitude of the momentum with which molecule moves away from the wall is again mυ x but in -x direction after the collision with the wall. Thus, net momentum change produced by wall The time interval between successive impacts is the time it takes for the molecule to travel to the wall opposite wall A and back again. If dimension of the cubic container is l

2 Then, distance l The time it takes to go this x -direction is l/ x momentum change per impact The average rate of change of momentum Time interval between successive impacts l / x l l Now, the pressure is he force per unit area / l l l l volume of the container This is a pressure P required to confine one molecule to a cubic container of volume. Molecular Basis of Gas Pressure- N molecules: Now, for N molecules If all molecules of gas has same mass N ( mυ x ) av N m ( x ) av

3 N where x ( x ) av P N () It is more convenient to express speed of molecule in three orthogonal components rather than just x component. x + y + z But for large number of molecules moving in random directions Thus From () and () we can write x y x x / z () P Nm () Problem The average speed of the nitrogen molecules in a sample of nitrogen gas at 5 0 C is 500 ms -. Show that the kinetic-molecular theory predicts that a pressure of about bar will be needed to confine -mol sample of nitrogen gas to a volume of 4.8 L at 5 0 C. P Nm 4.8 L m. υ 500 ms -. moles of N molecules Nm molar mass of N molecules 8 g 0.08 kg Nmυ (0.08 kg) (500 ms - ) ( m ) kg m s - / m kg m s - / m N / m

4 bar bar Problem a. Calculate the pressure needed to confine 0 gas particles, each of mass0-5 kg, to a -L container if the rms speed is 000 m s -. P Nm In this case L 0 - m, N 0 molecules, m 0-5 kg, 00 m s - Nm (0 ) (0-5 kg) (00 ms - ) (0 - m ) P and Kinetic Energy of Molecules: kg ms-/m N/m N kg ms - 0. bar bar 0 5 Nm - Average kinetic energy of a molecule of a gas, indicated by ke, is ke m m ke (4) From equation () and (4) P N(ke) P Nke (5) The number of molecules in the sample can be expressed in mole units by using a relation N n N A where n number of moles, N A Avogadro's constant (6.0 0 mol - ) P n N A ke

5 P n (N A ke ) If KE is used to denote translational energy of Avogadro's number of molecules, then P n KE (6) Thus, kinetic molecular theory showed us that product of volume of gas sample and pressure applied to confine sample is an energy quantity, which depends upon the number of gas molecules and translational energy of these molecules. Kinetic Energy and Temperature: From empirical ideal-gas law From (6) and (7) P nrt (7) n KE nrt KE RT (8) Thus, we can now postulate that: The translational kinetic energy of the molecules of -mol gas sample is equal to RT/.