Brownian Motion A Smoke Cell

The Gas Laws What variables are important here and what is their relationship?

Boyle’s Law pV=constant p = constant x 1/V Boyle's Law describes the relationship between the pressure and volume of a fixed mass of gas at constant temperature . The pressure is caused by the gas molecules bumping into the walls of the container. The pressure exerted by a gas at constant temperature varies inversely with the volume of the gas. pV=constant p = constant x 1/V

p1V1=p2V2 (at constant temperature) As pV = constant that means that p1V1 = constant and so does p2V2. We can therefore write: p1V1=p2V2 (at constant temperature) where V1 equals the original volume, V2 equals the new volume, p 1 the original pressure, and p2 the new pressure.   To get a straight line graph you need to rearrange the equation into the form Y = mx+c pV = constant so, p = constant x 1/V or, V = constant x 1/p There is no intercept - the line goes through the origin.

Charles’ Law V/T = constant Charles' Law describes the relationship between the volume and temperature of a fixed mass of gas that is held at a fixed pressure. As the temperature decreases the particles have to occupy a smaller volume if the pressure is to remain constant. V/T = constant V = constant x T A straight line is obtained with an intercept of -273.15 oC. This is true whatever the mass of the gas and it’s initial volume. If we use the Kelvin scale we can say that volume is proportional to the absolute temperature.

Pressure Law p/T = constant p = constant x T The pressure of a fixed mass of gas at constant volume is directly proportional to temperature (using the Kelvin scale) p/T = constant p = constant x T

When do you think they are not valid and why not? These 3 Laws are valid for all real gases to a high level of precision in most cases, when they behave like an “ideal gas”. When do you think they are not valid and why not? Ideal gases are gases that obey Boyle’s Law

The Ideal Gas Equation pV = nRT An Ideal Gas is a gas that obeys Boyle’s law. We can combine the 3 experimental gas laws: pV/T = constant for a fixed mass of ideal gas, using Kelvin. Or pV = constant x T This constant is the molar gas constant, R So pV = RT where R = 8.31J per K per mole (on data sheet) And for n moles of an ideal gas, pV = nRT This is the ideal gas equation (on data sheet) So the gradient nR of the line depends only on the amount of gas p1V1 = p2V2 T1 T2 1 mole of any ideal gas at 273K and 101kPa has a volume of 0.0224m3 So using pV/T we get a constant of 8.31Jmol-1K-1 Why joules?

pV against T k = R/NA R = kNA pV /Nm Gradient = pV/T NA is the Avogadro constant = the number of atoms in exactly 12g of carbon 12 isotope = 6.023 x 1023 1 mole contains NA particles n = number of moles in a substance or molarity (mol) So N = nNA = total number of particles in the substance pV against T pV /Nm Gradient = pV/T = N X m3 = Nm = J so gradient = JK-1 per mol m2 For n moles, the gradient = nR T/K Using pV = nRT, if we substitute n = N/NA and put in the Boltzmann constant k = R/NA R = kNA We get pV = NkT (data sheet) k is given on the data sheet, k = 1.38 x 10-23JK-1 pV = n R T pV = N x kNA x T NA

pV= (1/3)Nmcrms² This video goes through the pV= (1/3)Nmcrms² derivation with animations. It is very good especially if you watch it with the textbook and pause it to check you are following! Remember you need this derivation