# UH PHYS 1302 - Ch17 (16 pages)

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## Ch17

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ch 17 study guide

- Pages:
- 16
- School:
- University of Houston
- Course:
- Phys 1302 - Introductory to Physics II

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Chapter 17 Phases and Phase Changes 1 Ideal Gases An ideal gas is a gas in which intermolecular interactions are negligibly small Ideal gases don t really exist in nature but the behavior of real gases can be understood by studying the ideal case The equation of state for an ideal gas is of pressure P volume V number of molecules N and temperature T is P V N kT 1 where k 1 38 10 23 J K is the Boltzmann constant In Chapter 16 we saw that the temperature and pressure of a gas are linearly related with a temperature of absolute zero 0 K corresponding to a pressure of zero From the ideal gas equation we can see the constant of proportionality is N k V P N k V T for a fixed number of molecules in a fixed volume the pressure is linearly related to the temperature Similarly for a fixed volume and fixed temperature the pressure depends linearly on the number of molecules P kT V N Pumping more air molecules into an already inflated tire increases the pressure For a fixed number of molecules and a fixed temperature the pressure depends inversely on the volume P N kT 1 V You take a balloon that s pumped up and squeeze it as the volume decreases the pressure increases Remember pressure is force per unit area and is measured in pascals Pa 1Pa 1N m2 1kg ms 2 A helium filled balloon at 23 C has a volume of 0 025 m3 and is at a pressure of 1 8 105 Pa How many helium atoms are in the balloon L Whitehead 1 Phys 1302 23 C 296 15 K P V N kT PV N kT 1 8 105 Pa 0 025m3 N 1 38 10 23 J K 296 15K N 1 1 1024 molecules A mole is a unit used to measure the amount of a substance It is defined as the amount of a substance that contains as many molecules as there are atoms in 12 g of carbon 12 There are 6 022 1023 atoms in 12 g of carbon 14 NA 6 022 1023 is called Avagadro s number Thus 1mole 6 022 1023 molecules NA 6 022 1023 molecules mole 2 Let n be the number of moles in a gas then the number of molecules is N nNA If we plug this into the ideal gas equation P V nNA kT R NA k 6 022 1023 molecules mole 1 38 10 23 J K 8 31J mol K P V nRT 3 where R is called the universal gas constant One mole of any substance has the same number of molecules but one mole corresponds to a different mass for different substances One mole of helium atoms has a mass of 4 00260 g while one mole of copper atoms has a mass of 63 546 g The atomic or molecular mass M of a substance is the mass in grams of one mole of that substance The mass of an individual atom or molecule then is m M NA 1 1 1024 molecules is how many moles N nNA N n NA 1 1 1024 molecules 6 022 1023 molecules mole 1 8moles L Whitehead 2 Phys 1302 Boyle s Law The pressure of a gas varies inversely with volume as long as temperature and the number of molecules are head constant P V N kT constant which means the product of pressure and volume doesn t change Pi Vi Pf Vf 4 Each of the curves in the figure corresponds to a different temperature thus these curves are called isotherms Charles s Law The volume of a gas divided by its temperature is constant as long as the pressure and number of molecules are constant V Nk constant T P which means volume over temperature doesn t change Vi Vf Ti Tf 5 If the temperature of the helium inside the balloon in the example above increases to 40 C but the pressure remains constant what volume will the gas occupy The number of molecules isn t changing because we are not pumping air into or out L Whitehead 3 Phys 1302 of the balloon Thus we can use Charles s Law 40 C 313 15 K Vi Vf Ti Tf Vi Vf Tf Ti 0 025m3 Vf 313 15K 296 15K Vf 0 026m3 2 Kinetic Theory The kinetic theory of gases makes the connection between what s happening in a gas on the microscopic level to what we observe on the macroscopic level Suppose we have a gas made up of a collection of molecules moving about inside a container of volume V We make the following assumptions 1 The container holds a very large number N of identical molecules each of mass m We assume the size of the molecules is negligible compared to the size of the container and compared to the distance between the molecules 2 The molecules move in a random manner and they obey Newton s law of motion 3 When molecules hit the walls of the container or collide with one another they bounce elastically Other than these elastic collisions the molecules have no interactions The pressure of a gas is due to the collisions between gas molecules and the walls of the container Suppose we have a container that is a cube with side length L Consider a molecule moving in the negative x direction toward a wall of the container Its speed is vx and thus its initial momentum is pi mvx When it hits the wall it bounces off elastically meaning that its momentum is now the same magnitude but in the opposite direction pf mvx The change in the momentum is p mvx mvx 2mvx The time it takes for the molecule to travel from one wall to the another wall bounce off and get back to the original wall is t 2L vx The average force is then F L Whitehead 2mvx mvx2 p t 2L vx L 4 Phys 1302 and the average pressure is P F mvx2 1 mvx2 A L L2 V 6 The molecules in a gas will have different speeds which are constantly changing as each molecule undergoes collisions with the walls and other moelcules But the distribution of speeds remains constant The Maxwell speed distribution named after James Clerk Maxwell is the probability that a molecule in a gas will have a particular speed The distribution depends on the temperature shown for O2 gas in the figure The most probable speed increases with temperature To make the pressure equation more general we replace vx2 with the average value vx2 avg P m vx2 avg V The above is the pressure exerted by one molecule To get the total pressure we multiply by the number of molecules P N m vx2 avg V This is only for velocity in the x direction The total average velocity will come from summing the 3 components v 2 avg vx2 avg vy2 avg vz2 avg 3 vx2 avg or vx2 avg 1 3 v 2 avg There s …

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