Gas Laws AP Physics B

The Periodic Table All of the elements on the periodic table are referred to in terms of their atomic mass. The symbol u is denoted as an atomic mass unit. 1 amu = 1.66 x kg

The Mole The word “mole” is derived from Latin to mean “heap” or “pile”. It is basically a chemical counting unit. Below, what we have is called the Mole Road Map and it summarizes what a mole equals. 1 MOLE

Example A flexible container of Oxygen(O 2 ) has a volume of 10.0 m 3 at STP. Find the # moles and molecules that exist in the container

Factors that effect a GAS 1. The quantity of a gas, n, in moles 2. The temperature of a gas, T, in Kelvin (Celsius degrees + 273) 3. The pressure of a gas, P, in pascals 4. The volume of a gas, V, in cubic meters

Gas Law #1 – Boyles’ Law (complete TREE MAP) “The pressure of a gas is inverse related to the volume” Moles and Temperature are constant

Gas Law #2 – Charles’ Law “The volume of a gas is directly related to the temperature” Pressure and Moles are constant

Gas Law #3 – Gay-Lussac’s Law “The pressure of a gas is directly related to the temperature” Moles and Volume are constant

Gas Law #4 – Avogadro’s Law “The volume of a gas is directly related to the # of moles of a gas” Pressure and Temperature are constant

Gas Law #5 – The Combined Gas Law You basically take Boyle’s, Charles’ and Gay-Lussac’s Law and combine them together. Moles are constant

Example Pure helium gas is admitted into a leak proof cylinder containing a movable piston. The initial volume, pressure, and temperature of the gas are 15 L, 2.0 atm, and 300 K. If the volume is decreased to 12 L and the pressure increased to 3.5 atm, find the final temperature of the gas.

Gas Law #6 – The IDEAL Gas Law All factors contribute! In the previous examples, the constant, k, represented a specific factor(s) that were constant. That is NOT the case here, so we need a NEW constant. This is called, R, the universal gas constant.

Example A helium party balloon, assumed to be a perfect sphere, has a radius of 18.0 cm. At room temperature, (20 C), its internal pressure is 1.05 atm. Find the number of moles of helium in the balloon and the mass of helium needed to inflate the balloon to these values.

So we have: The absolute pressure P of an ideal gas is directly proportional to the Kelvin temperature T and the number of moles n of the gas and is inversely proportional to the volume V of the gas:

Kinetic Theory of Gases 1. The container holds a very large number N of identical molecules. Each molecule has a mass m, and behaves as a point particle. 2. The molecules move about the container in a random manner. They obey Newton’s laws of motion at all times. 3. When the molecules hit the walls of the container or collide with one another, they bounce elastically. Other than these collisions, the molecules have no interactions.

For conditions of low gas density, the distribution of speeds of the particles within a large collection of molecules at constant temperature was calculated by Maxwell. See the figure below for Maxwellian distribution curve.

To find the force exerted by the molecule impacts on the container wall consider: ideal gas of N identical particles in a cubical container whose sides have length L. Except for elastic collisions (where on average there is no gain or loss of translational K) the particles do not interact focus on one particle of mass m as it strikes a wall perpendicularly and rebounds elastically.  while approaching the wall, it has a velocity +v and linear momentum +mv  the particle rebounds with a velocity –v and linear momentum –mv, travels to the opposite wall, and rebounds again  the time t between collisions with the first wall is the round-trip distance 2L divided by the speed of the particle: t = 2L / v  Impulse-momentum theorem says the force exerted on the particle by the wall is equal to the change in momentum of the particle per unit time:

Cont.  Newton’s 3rd law tells us that this force (of wall-on-particle) is equal and opposite to the force of the particle on the wall. Thus, F= + mv2/L.  the total force exerted on the wall is equal to the number of particles times the average force for each particle.  Since N particles move randomly in 3-D, about 1/3 of them will strike the wall, that is N/3. Therefore the total force is:  Notice v2 has been replaced with, the average squared speed. (remember Maxwellian distribution). This quantity is called the root- mean-square speed, or rms speed;. Thus:

Cont.  Pressure is F/A, so P acting on a wall of area L2 is  Volume of the box is V = L3, so:  Of course, is the average translational kinetic energy :  This is similar to PV= N k T. set these equal (PV = PV):  This can be rearranged:  Thus,. Rearranging yields:

Example The atmosphere is composed primarily of nitrogen N 2 (78%) and oxygen O 2 (21%). (a) Is the rms speed of N 2 (28.0 g/mol) greater than, less than, or the same as the rms speed of O 2 (32.0 g/mol)? (b) Find the rms speed of N 2 and O 2 at 293 K.

The Internal Energy of a Monatomic Ideal Gas Internal Energy – sum of the various kinds of energy that the atoms or molecules of the substance possess. The only energy contributing to internal energy of a monatomic ideal gas is translational kinetic energy. Thus, the total internal energy, U is the number of particles times the kinetic energy of each particle: U = N (). Using the equation derived above, this becomes: Remember that k = R / NA and N / NA = n