Temperature, Thermal Expansion and the Gas Laws

Transcription

1 Temperature, Thermal Expansion and the Gas Laws z x Physics 053 Lecture Notes

2 Temperature,Thermal Expansion and the Gas Laws Temperature and Thermometers Thermal Equilibrium Thermal Expansion The Ideal Gas Law Molecular Interpretation of Temperature

3 Temperature and Thermometers Temperature is a measure of how hot or cold something is. Thermometers are instruments designed to measure temperature. In order to do this, they take advantage of some property of matter that changes with temperature. Most materials expand when heated.

4 Temperature and Thermometers Common thermometers used today include the liquid-in-glass type and the bimetallic strip.

5 Temperature and Thermometers Temperature is generally measured using either the Kelvin, Celsius, or the Fahrenheit scale. Boiling Point (H O) Melting Point (H O) Kelvin Celsius Fahrenheit Absolute Zero

6 Thermal Equilibrium Two objects placed in thermal contact will eventually come to the same temperature. When they do, we say they are in thermal equilibrium. The Zeroth Law of Thermodynamics C A B If A is in thermal equilibrium with C and B is in thermal equilibrium with C Then A and B are in thermal equilibrium.

7 Thermal Expansion Expansion occurs when an object is heated. A steel washer is heated Does the hole increase or decrease in size?

8 Thermal Expansion When the washer is heated The hole becomes larger

9 Thermal Expansion L o L L L o T T L αl o T L L o αl o T Coefficient of linear expansion L L o (1+α T)

10 Expansion Problem Thermal Expansion shaft sleeve D d A cylindrical brass sleeve is to be shrunk-fitted over a brass shaft whose diameter is 3.1 cm at 0 o C. The diameter of the sleeve is cm at 0 o C. To what temperature must the sleeve be heated before it will slip over the shaft?

11 Expansion Problem Thermal Expansion To what temperature must the sleeve be heated before it will slip over the shaft? shaft D sleeve d L D d L α L i D d αd T ( ) T f T i α 19 x / o D 3.1 cm d cm T 0oC i C D d αdt f T f ( D d) αd ( 3.1 cm cm) 6 1/ o C( cm) 63 o C 19 x 10

12 Volume Expansion Thermal Expansion L o + L L o L o Lo L o + L L o + L New Volume V ( L ) 3 o + L Initial Volume V o L 3 o V L V V V 3 ( L) + 3 o + 3L o L + 3Lo L V V o 3L o L 3L o ( αlo T) 3 3αL o T L αl o T V 3αVo T

13 Thermal Expansion V 3αVo T Coefficient of Volume Expansion β 3α V βvo T V V o βv o T ( 1 + β T) V Vo

14 Expansion Problem An automobile fuel tank is filled to the brim with 45 L of gasoline at 10 o C. Immediately afterward, the vehicle is parked in the Sun, where the temperature is 35 o C. How much gasoline overflows from the tank as a result of expansion? Overflow Change in volume of the gasoline Thermal Expansion V V V G V S V β V G V ( β β ) V T G o T β S ( 4 ) 9.6 x x 10 45( 5) V 1.04 L o S V o T Change in volume of the steel gas tank

15 Thermal Expansion Potential Energy High Temperature Low Temperature Intermolecular Distance

16 The Ideal Gas Law Pressure (Pa) Volume (m 3 ) PV nrt Absolute Temperature (K) Gas Quantity (mol) Gas Constant (8.31 J/mol. K) (L. atm)/(mol. K) 1.99 cal/(mol. K)

17 The Ideal Gas Law A mole (mol) is defined as the number of grams of a substance that is numerically equal to the molecular mass of the substance: 1 mol H has a mass of g 1 mol Ne has a mass of 0 g 1 mol CO has a mass of 44 g The number of moles in a certain mass of material: mass n ( mol) molecular mass ( grams) ( g / mol) n m Μ The number of moles in a certain number of particles: ( mol) n molecules (particles) Avogadro's number ( particles / mol) n N N A

18 The Ideal Gas Law Pressure (Pa) Volume (m 3 ) PV NkT Absolute Temperature (K) Number of Molecules Boltzmann s Constant (1.38 x 10-3 J/K)

19 The Ideal Gas Law Boltzmann s Constant PV nrt NkT k nr N Gas Constant k R N n Avogadro s Number

20 The Ideal Gas Law A gas is contained in an 8.0 x 10 3 m 3 vessel at 0 o C and a pressure of 9.0 x 10 5 N/m. (a) Determine the number of moles of gas in the vessel. PV nrt 5 ( 3 ) 9.0 x 10 N/m 8.0 x 10 m n PV RT 8.31 J / mol K ( 93 K) n 3.0 mol

21 The Ideal Gas Law A gas is contained in an 8.0 x 10 3 m 3 vessel at 0 oc and a pressure of 9.0 x 10 5 N/m. n 3.0 mol (b) How many molecules are in the vessel? N nn A 3.0 mol ( 3 molecules) 6.0 x 10 mol N 1.8 x 10 4 molecules

22 The Ideal Gas Law Problem A cylinder with a moveable piston contains gas at a temperature of 7 o C, a volume of 1.5 m 3, and an absolute pressure of 0.0 x 10 5 Pa. 0.0 x 10 5 Pa 1.5 m 3 7 o C 0.80 x 10 5 Pa 0.70 m 3 What will be its final temperature if the gas is compressed to 0.70 m 3 and the absolute pressure increases to 0.80 x 10 5 Pa?

23 Problem 0.0 x 10 5 Pa What will be its final temperature if the gas is compressed to 0.70 m 3 and the absolute pressure increases to 0.80 x 10 5 Pa? 1.5 m 3 PV Pf V T f nrt Pi V T i P f V Pi V PV T The Ideal Gas Law nr constant f i ( 5 ) f 0.8 x 10 Pa T f Ti 300 K 5 i 0. x 10 Pa 7 o C 0.7 m 560 K ( ) m x 10 5 Pa m 3 87 o C

24 Molecular Interpretation of Temperature Assumptions of kinetic theory: 1) large number of molecules, moving in random directions with a variety of speeds ) molecules are far apart, on average 3) molecules obey laws of classical mechanics and interact only when colliding 4) collisions are perfectly elastic

25 Molecular Interpretation of Temperature y z L v A x The force exerted on the wall by the collision of one molecule of mass m is F ( mv) t mv L v x x mv x L Then the average force due to N molecules colliding with that wall is F m L Nv x

26 Molecular Interpretation of Temperature z L y v A x The averages of the squares of the speeds in all three directions are equal: F 1 3 mnv L F m L Nv x So the pressure on the wall is: P F A 1 3 Nmv AL 1 3 Nmv V

27 Molecular Interpretation of Temperature P 1 3 Nmv V Rewriting, PV ( 1 m ) N 3 v NkT so ( 1 mv ) kt 3 ( KE) 1 mv 3 kt The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas.

28 Molecular Interpretation of Temperature Molecular Kinetic Energy Temperature is a measure of the average molecular kinetic energy. mv KE mv 3kT 3kT v( rms ) 3kT m

29 Molecular Interpretation of Temperature Problem What is the total random kinetic energy of all the molecules in one mole of hydrogen at a temperature of 300 K. Avogadro s number Kinetic energy per molecule K 3 NA kt K 6.0 x 10 3 molecules 1.38 x J K ( 300 K) K 3740 J

30 Molecular Interpretation of Temperature Problem Calculate the rms speed of a Nitrogen molecule (N ) when the temperature is 100 o C. Mass of N molecule: m 8.0 x x kg/mole molecules/mole x 10 kg rms speed: v( rms ) 3kT m ( 3 J/K) x x 10 6 kg 373 K v( rms ) 576 m/s

31 Molecular Interpretation of Temperature Problem If.0 mol of an ideal gas are confined to a 5.0 L vessel at a pressure of 8.0 x 10 5 Pa, what is the average kinetic energy of a gas molecule? Temperature of the gas: PV nrt ( 3 3 ) m PV 8.0 x 10 Pa 5.0 x 10 T nr J.0 mol 8.31 mol K Kinetic energy: K 3 kt x 10 J K 44 K 44 K 1.38 x 10 3 J molecule

32 Molecular Interpretation of Temperature Mean and rms Speed Mean Speed: v( mean ) 3.6 m/s rms Speed: ( 1) + ( 6) + ( 4) + ( ) + ( 6) + ( 3) + ( ) + ( 5) 8 v( rms ) 4.0 m/s

33 Summary Temperature is a measure of how hot or cold something is, and is measured by thermometers. There are three temperature scales in use: Celsius, Fahrenheit, and Kelvin. When heated, a solid will get longer by a fraction given by the coefficient of linear expansion. The fractional change in volume of gases, liquids, and solids is given by the coefficient of volume expansion.

34 Summary Ideal gas law: PV nrt One mole of a substance is the number of grams equal to the atomic or molecular mass. Each mole contains Avogadro s number of atoms or molecules. The average kinetic energy of molecules in a gas is proportional to the temperature: ( KE) 1 mv 3 kt

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