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GAS LAWS AND THERMAL PROPERTIES OF MATTER PV=nRT

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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

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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 N A =6.022x10 23 atoms of 1 element

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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)

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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

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Example- Mass of air in a Scuba Tank A scuba tank usually has a volume of 11.0L a gauge pressure of 2.10x10 7 Pa when full. The “empty” tank contains 11.0L of air at 21.0 o C at 1 atm. A compressor pumped air into the tank, increasing its temperature to 42.0 o C and the pressure to 2.11x10 7 Pa. What mass of air was added? (air is 78% N, 21% O, 1% misc. Average molar mass is 28.8g/mol)

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Example

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Example – Young and Freedman 18.3 A cylindrical tank has a tight fitting piston that allows its volume to change. The tank originally contains 0.110m 3 of air at a pressure of 3.40atm. The piston is slowly pulled out until the tank has a volume of 0.390m 3. If temperature remains constant what is the final value of the pressure?

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PV=nRT=constant

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Boltzmann’s Constant When dealing with macroscopic variables Sometimes instead of n, number of moles, you have N, number of molecules Where k B is boltzmann’s constant Look at my magnificent beard

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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

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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

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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

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Kinetic Energy of Gas That is the force of 1 molecule Total force is will be Where N is the number of molecules

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Kinetic Energy of Gas Average velocity squared is Total force is We can generalize this for velocity in all directions

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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

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Kinetic Energy of Gas Rearrange terms Divide by Area

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Rearrange Terms Holds to some extent for liquids and even solids

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Root Mean Squared Speed Average of velocities squared (not the square of the average velocities)

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Example – Young and Freedman Martian Climate- Atmosphere of Mars is mostly CO 2 (M=44.0g/mol) under a pressure of 650 Pa, which we shall assume remains constant. The temperature varies from 0.0 o C to -100 o C througout the year. Over the course of a year, what are the ranges of (a) the rms speeds of the CO 2 molecules (b) the density in mol/m 3 of the atmosphere?

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Example 0 o C -100 o C

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Example Density in mol/m 3 0 o C 100 o C

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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

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Phase Diagram of Water

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Phase Diagram of CO 2

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Phase Diagram Triple Point- the only Pressure and Temperature where all three phases can exist Critical Point – Differences in solid and liquid phase disappear

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PVT Surfaces

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Example – Young and Freedman 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=T 0 - α y where T0=15oC and α=6.0 o C/1000m

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Giancoli A helium filled balloon is escapes a child's hand at sea level and 20.0 o C. When it reaches an altitude of 3000m where the temperature is 5.00 o C and the pressure is only atm, how will its volume compare to that at sea level?

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Young and Freedman With the assumption that the air temperature is a uniform 0.0 o C, what is the density of the air at an altitude of 1.00km as a percentage of the density at the surface?

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Giancoli Two isotopes for uranium are 235 U and 238 U. They can be separated by gas diffusion by combining them with fluorine to form UF 6. Calculate the ratio of the RMS speeds of these molecules for the two isotopes at constant T.

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