Equations of State • State Variables – Variables that describe the condition or state a material is in (macroscopic) • Ex. Volume, Pressure, Temperature • Equations of State – shows the relationship of different state variables • Can get complicated but made simpler through approximations • Ex. Equations of state for a solid
Ideal Gas • An ideal gas is composed of randomly moving, non-interacting point particles. • Difficult to describe gas in terms of mass • Easier to use • Where n is the number of moles and • M is the molar mass (mass per mole of gas) • For those who’ve forgotten one mole is NA=6.022x1023 atoms of 1 element
Ideal Gas • Volume is proportional to the number of moles of the gas (keeping Pressure and temperature the same) • Volume is inversely proportional to pressure (keeping number of moles and temperature the same) • Pressure is proportional to the absolute temperature (keeping volume and number of moles the same)
Ideal Gas Equation • Where R is the gas constant R=8.314J/molK • This equation will work for an ideal gas for all pressures and temperatures • For real world gasses this still holds for low pressures and temperatures • Its off by a few percent at higher P • Note how PV = energy
Example- Mass of air in a Scuba Tank • A scuba tank usually has a volume of 11.0L a gauge pressure of 2.10x107 Pa when full. The “empty” tank contains 11.0L of air at 21.0oC at 1 atm. A compressor pumped air into the tank, increasing its temperature to 42.0oC and the pressure to 2.11x107Pa. What mass of air was added? (air is 78% N, 21% O, 1% misc. Average molar mass is 28.8g/mol)
Example – Young and Freedman 18.3 • A cylindrical tank has a tight fitting piston that allows its volume to change. The tank originally contains 0.110m3 of air at a pressure of 3.40atm. • The piston is slowly pulled out until the tank has a volume of 0.390m3. If temperature remains constant what is the final value of the pressure?
Boltzmann’s Constant • When dealing with macroscopic variables • Sometimes instead of n, number of moles, you have N, number of molecules • Where kB is boltzmann’s constant • Look at my magnificent beard
Van der Waals Equation • Corrects some omissions in ideal gas equation • Where b is the volume of 1 mole of gas and a is the attractive forces between the molecules • If n/V is small, the gas is dilute and it reduces to the ideal gas equation
Kinetic Energy of a gas • Gas molecules are always in motion • Kinetic energy of molecules proportional to its temperature • That means gas in a container also has energy • Assumptions • Volume V has large number N of identical molecules • Molecules are point particles • Molecules are in constant motion and collide elastically • Container walls are rigid and do not move
Kinetic Energy of Gas • Collision of molecules with container is what causes pressure. • CHEAT: instead of using time the molecule is in contact with the wall, we can use the time it takes for the molecule to impact the wall again • Molecule needs to travel 2d where d is the side of the container
Kinetic Energy of Gas • That is the force of 1 molecule • Total force is will be • Where N is the number of molecules
Kinetic Energy of Gas • Average velocity squared is • Total force is • We can generalize this for velocity in all directions
Kinetic Energy of Gas • Average velocity squared will equal (Pythagorean for 3d) • Somewhat cheat: because gas molecules move randomly, the probability of moving in any direction is the same. Average velocity in one direction will equal average velocity in all directions
Kinetic Energy of Gas • Rearrange terms • Divide by Area
Rearrange Terms • Holds to some extent for liquids and even solids
Root Mean Squared Speed • Average of velocities squared (not the square of the average velocities)
Example – Young and Freedman 18.36 • Martian Climate- Atmosphere of Mars is mostly CO2 (M=44.0g/mol) under a pressure of 650 Pa, which we shall assume remains constant. The temperature varies from 0.0oC to -100oC througout the year. Over the course of a year, what are the ranges of (a) the rms speeds of the CO2 molecules (b) the density in mol/m3 of the atmosphere?
Example • 0oC • -100oC
Example • Density in mol/m3 • 0oC • 100oC
Phase Changes of Matter • We now relate phase changes with pressure as well • A change in phase will occur when the system is in phase equilibrium • one is to one relationship between pressure and temperature • We can plot PvsT on a graph to create a phase diagram
Phase Diagram • Triple Point- the only Pressure and Temperature where all three phases can exist • Critical Point – Differences in solid and liquid phase disappear
Example – Young and Freedman 18.50 • Puffy Cumulous clouds which are made of water droplets, occur at lower altitudes of the atmosphere. Wispy Cirrus clouds, made of ice droplets, only occur at higher altitudes. Find the Altitude y above which only cirrus clouds can occur. On a typical day and an altitude less than 11 km, the temperature at any given altitude is T=T0-αy where T0=15oC and α=6.0oC/1000m
Giancoli 13-40 • A helium filled balloon is escapes a child's hand at sea level and 20.0oC. When it reaches an altitude of 3000m where the temperature is 5.00oC and the pressure is only 0.700 atm, how will its volume compare to that at sea level?
Young and Freedman 18.20 • With the assumption that the air temperature is a uniform 0.0oC, what is the density of the air at an altitude of 1.00km as a percentage of the density at the surface?
Giancoli 13-60 • Two isotopes for uranium are 235U and 238U. They can be separated by gas diffusion by combining them with fluorine to form UF6. Calculate the ratio of the RMS speeds of these molecules for the two isotopes at constant T.