Physics 231 Topic 12: Temperature, Thermal Expansion, and Ideal Gases Alex Brown Nov 18-23 2015

homework 3rd midterm final Thursday 8-10 pm makeup Friday final 9-11 am

Key Concepts: Temperature, Thermal Expansion, and Ideal Gases Temperature and Thermometers Thermal Energy & Temperature Thermal Expansion Coefficient of thermal expansion Ideal Gases State Variables Ideal gas law Kinetic Theory of Gases Kinetic & thermal energy Maxwell distribution Covers chapter 12 in Rex & Wolfson

Conversions: Tc = Tk - 273.15 Tf = (9/5)Tc + 32 Helium boils at Tk=4

Binding Forces Kinetic energy ~ T (temperature) Potential Energy -Emin -Emin R The curve depends on the material, e.g. Emin is different for water and iron R 2 atom/molecules

Solid (low T) Kinetic energy ~ T Potential Energy Rmin R -Emin R -Emin The temperature (and thus kinetic energy) is so small that the atoms/molecules can only oscillate around a fixed position Rmin

Liquid (medium T) Potential Energy Kinetic energy ~ T Rmin R -Emin R -Emin On average, the atoms/molecules like to stick together but sometimes escape and can travel far.

Gas (high T) Kinetic energy ~ T Potential Energy Rmin -Emin R -Emin R The kinetic energy is much larger than Emin and the atoms/molecules move around randomly.

What happens if the temperature of a substance is increased? Rmin=Rave(T=0) Kinetic energy ~ T R T=0: Average distance between atoms/molecules: Rmin -Emin

What happens if the temperature of a substance is increased? Rmin=Rave(T=0) Kinetic energy ~ T Rave(T>0) > Rmin T>To: The average distance between atoms/molecules is larger than Rmin: the substance expands R -Emin

Thermal expansion L= Lo T A =  Ao T  = 2 V =  Vo T  = 3 length L L= Lo T surface A =  Ao T  = 2 volume V =  Vo T  = 3 L0 : coefficient of linear expansion different for each material Some examples:  = 24x10-6 1/K Aluminum  = 1.2x10-4 1/K Alcohol T=T0 T=T0+T

A Heated Ring A metal ring is heated. What is true: The inside and outside radii become larger The inside radius becomes larger, the outside radius becomes smaller The inside radius becomes smaller, the outside radius becomes larger The inside and outside radii become smaller PHY 231

Demo: Bimetallic Strips top bottom Application: contact in a refrigerator top<bottom if the temperature increases, The strip curls upward, makes contact and switches on the cooling.

Water: a special case Coef. of expansion is negative: If T drops the volume becomes larger Coef. Of expansion is positive: if T drops the volume becomes smaller Below this ice is formed (it floats on water)

Ice  (g/cm3) liquid 1 Phase transformation 0.917 ice Ice takes a larger volume than water! A frozen bottle of water might explode

Thermal equilibrium Thermal contact High temperature Low temperature High kinetic energy Particles move fast Low temperature Low kinetic energy Particles move slowly Transfer of kinetic energy Thermal equilibrium: temperature is the same everywhere

Zeroth law of thermodynamics If objects A and B are both in thermal equilibrium with an object C, than A and B are also in thermal equilibrium. There is no transfer of energy between A, B and C

Ideal Gas: properties Collection of atoms/molecules that Exert no force upon each other The energy of a system of two atoms/molecules cannot be reduced by bringing them close to each other Take no volume The volume taken by the atoms/molecules is negligible compared to the volume they are sitting in

Potential Energy Rmin -Emin R Ideal gas: we are neglecting the potential energy between The atoms/molecules Potential Energy Kinetic energy R

Properties of gases V = volume P = pressure T = temperature in K (Kelvin) n = number of moles Example balloon

Molecular mass

Name Number of electrons Z X A molar mass in grams

Weight of 1 mol of atoms 1 mol of atoms weighs A grams (A is the molar mass) Examples: 1 mol of Hydrogen weighs 1.0 g 1 mol of Carbon weighs 12.0 g 1 mol of Oxygen weighs 16.0 g 1 mol of Zinc weighs 65.4 g What about molecules? H2O 1 mol of water molecules: 2 x 1.0 g (due to Hydrogen) 1 x 16.0 g (due to Oxygen) Total: 18.0 g

Number of atoms and moles

Example A cube of Silicon (molar mass 28.1 g) is 250 g. How much Silicon atoms are in the cube? B) What would be the mass for the same number of gold atoms (molar mass 197 g) Total number of moles n = M / Mmolar = 250/28.1 = 8.90 N = n NA = (8.9) (6.02x1023) = 5.4x1024 atoms M = n Mmolar = (8.90) (197 g) = 1750 g

Question 1 mol of CO2 has a larger mass than 1 mol of CH2 1 mol of CO2 contains more molecules than 1 mol of CH2 1) true 2) true 1) true 2) false 1) false 2) true 1) false 2) false

Properties of gases V = volume P = pressure T = temperature in K (Kelvin) n = number of moles Example balloon

Boyle’s Law (fixed n and T) ½P0 2V0 P0 V0 2P0 ½V0 At constant temperature: P ~ 1/V implies that PV = constant

Charles’ law (fixed n and P) 2V0 2T0 V0 T0 If you want to maintain a constant pressure, the temperature must be increased linearly with the volume V ~ T implies that (V/T) = constant

Gay-Lussac’s law (fixed n and V) P0 T0 2P0 2T0 If, at constant volume, the temperature is increased, the pressure will increase by the same factor P ~ T implies that (P/T) = constant

Brown’s law (fixed T and P) 2n0 2V0 n0 V0 If you double the number of particles the volume doubles n ~ V implies that (V/n) = constant

Boyle & Charles & Gay-Lussac IDEAL GAS LAW Does not depend on what type or atom or molecule n = number of moles R = universal gas constant 8.31 J/mol·K If the number of moles is fixed

Example An ideal gas occupies a volume of 1.0 cm3 at 200 C at 1 atm. How many atoms are in the volume? B) If the pressure is reduced to 1.0x10-11 Pa, while the temperature drops to 00C, how many atoms remained in the volume? PV = nRT, so n = PV/(TR) with R=8.31 J/mol K T=200C=293K, P=1atm=1.013x105 Pa, V=1.0cm3=1x10-6m3 n=4.2x10-5 mol N = n NA = (4.2x10-5) NA=2.5x1019 T = 00C = 273K , P = 1.0x10-11 Pa, V = 1x10-6 m3 n=4.4x10-21 mol N=2.6x103 particles (almost vacuum)

And another! An air bubble has a volume of 1.50 cm3 at 950 m depth (T=7oC). What is its volume when it reaches the surface (T=20oC). (water=1.0x103 kg/m3)? P950m=P0 + water g h = 1.013 x 105 + (1.0x103)(9.8)(950) = 94.2 x 105 Pa Vsurface=146 cm3 Expanded by a factor of 97

Quiz A volume with dimensions L x W x H is kept under pressure P at temperature T. If the temperature is raised by a factor of 2, and the height is made 5 times smaller, by what factor does the pressure change, i.e. what is P2/P1? No gas leaks or is added. a) 0.4 b) 1 c) 2.5 d) 5 e) 10 Use the fact PV/T is constant if no gas is added/leaked P1V1 / T1 = P2V2 / T2 P1V1 / T1 = P2 (V1/5) / (2T1) P2 = (5)(2)(P1 ) = 10 P1 a factor of 10.

“Standard temperature and pressure” (STP)

Moles

macroscopic to microscopic macroscopic quantities N = number of atoms or molecules (microscopic)

Quiz Given P1 = 1 atm P2 = 2 atm V1 = 2 m3 V2 = 10 m3 T1 = 100 K N1 = NA N2 = 10 NA T2 = ? K 200 500 2000 5000 100

Example How many air molecules at in the room with a volume of 1000 m3 (assume only molecular nitrogen is present N2)? PV = N kB T T = 293 P = 1.013x105 Pa V = 1000 m3 N = 2.5x1028

microscopic description: kinetic theory of gases 1) The number of objects is large (statistical model) 2) Their average separation is large 3) The objects follow Newton’s laws 4) Any particular object can move in any direction with a distribution of velocities 5) The objects undergo elastic collision with each other 6) The objects make elastic collisions with the walls 7) All objects are of the same type

Movie of gas in two dimensions

mean free path d = average distance between collisions air at P = 1 atm d = 68 nm = 68 x 10-9 m high vacuum P = 10-5 Pa d = 1m in space P = 10-12 Pa d = 108 m

The Maxwell Distribution However we can model the distribution of the velocities (& thus the kinetic energies) of the individual gas molecules. The result is the Maxwell Distribution. The root-mean-square (rms) velocity is

Energy of one object Objects inside the container have a distribution of velocities around an average – so each object has an average kinetic energy given by average translation kinetic energy average squared velocity mass of the object (atom or molecule)

Relationship to ideal gas law The objects bounce off of each other and the walls of the container (elastic). One can derive the following result How the average kinetic energy of one atom is related to temperature

root-mean-square (rms) velocity for one atom or molecule

Example What is the rms speed of air at 1 atm and room temperature (293 K)? Assume it consist of molecular Nitrogen only (N2)? R = 8.31 J/mol K T = 293 K Mmolar = (2 x 14)x10-3 kg/mol vrms = 511 m/s = 1140 mph !

Total thermal energy d is the number of “degrees of freedom” for the motion d = 3 for an atom (motion in x, y, z directions) like helium gas d = 5 for a diatomic molecule (motion in x, y, z and two ways to rotate) like nitrogen molecule N2 or hydrogen molecule H2 (Homework question for “one degree of freedom” use d = 1)

Example What is the total thermal kinetic energy of the air molecules in the lecture room (assume only molecular nitrogen is present N2)? Eth = (d/2) PV = 2.5x108 J d = 5 P = 1.013x105 Pa V = 1000 m3 Using KE = (1/2) mv2 this is equivalent to 1000 cars with m=1000 kg each moving with v = 22.3 m/s (50 mph) Can we use that energy to do work?

Diffusion