Chapter 13: Temperature and Ideal Gas

Transcription

1 Chapter 13: Temperature and Ideal Gas What is Temperature? Temperature Scales Thermal Expansion Molecular Picture of a Gas The Ideal Gas Law Kinetic Theory of Ideal Gases Chemical Reaction Rates Collisions Between Molecules 1

2 13.1 Temperature Heat is the flow of energy due to a temperature difference. Heat always flows from objects at high temperature to objects at low temperature. When two objects have the same temperature, they are in thermal equilibrium.

3 The Zeroth Law of Thermodynamics: If two objects are each in thermal equilibrium with a third object, then the two objects are in thermal equilibrium with each other. 3

4 13. Temperature Scales Absolute or Kelvin scale Fahrenheit scale Celsius scale Water boils * K 1 F 100 C Water freezes * K 3 F 0 C Absolute zero 0 K F C (*) Values given at 1 atmosphere of pressure. 4

5 The temperature scales are related by: Fahrenheit/ Celsius T (.8 F/ C) + 3 F F 1 TC Absolute/ Celsius T T C

6 Example (text problem 13.3): (a) At what temperature (if any) does the numerical value of Celsius degrees equal the numerical value of Fahrenheit degrees? T F 1.8T T C C C T C (b) At what temperature (if any) does the numerical value of Kelvin equal the numerical value of Fahrenheit degrees? T T F F 1.8T C + 3 ( T 73) ( T 73) F 574 F

7 13.3 Thermal Expansion of Solids and Liquids Most objects expand when their temperature increases. 7

8 An object s length after its temperature has changed is ( 1+ ) L 0 L αδt α is the coefficient of thermal expansion where ΔTT-T 0 and L 0 is the length of the object at a temperature T 0. 8

9 How does the area of an object change when its temperature changes? The blue square has an area of L 0. L 0 L 0 +ΔL With a temperature change ΔT each side of the square will have a length change of ΔL αδtl 0. 9

10 10 ( )( ) ( ) T A A T A TL L L T TL L TL L TL L Δ Δ Δ + Δ + Δ + Δ + Δ + Δ + α α α α α α α 1 A new area The fractional change in area is:

11 The fractional change in volume due to a temperature change is: ΔV V 0 βδt For solids β3α 11

12 13.4 Molecular Picture of a Gas The number density of particles is N/V where N is the total number of particles contained in a volume V. If a sample contains a single element, the number of particles in the sample is N M/m. N is the total mass of the sample (M) divided by the mass per particle (m). 1

13 One mole of a substance contains the same number of particles as there are atoms in 1 grams of 1 C. The number of atoms in 1 grams of 1 C is Avogadro s number. N A 3 mol -1 13

14 A carbon-1 atom by definition has a mass of exactly 1 atomic mass units (1 u). This is the conversion factor between the atomic mass unit and kg (1 u kg). N A and the mole are defined so that a 1 gram sample of a substance with an atomic mass of 1 u contains exactly N A particles. 14

15 Example (text problem 13.39): Air at room temperature and atmospheric pressure has a mass density of 1. kg/m 3. The average molecular mass of air is 9.0 u. How many air molecules are there in 1.0 cm 3 of air? number of particles 3 total mass of air in 1.0 cm average mass per air molecule The total mass of air in the given volume is: m ρv 1. kg 3 m cm 1 6 kg 3 1m 100 cm 3 15

16 Example continued: number of particles 3 total mass of air in 1.0 cm average mass per air molecule kg ( )( u/particle kg/u) particles 16

17 13.5 Absolute Temperature and the Ideal Gas Law Experiments done on dilute gases (a gas where interactions between molecules can be ignored) show that: For constant pressure V T Charles Law For constant volume P T Gay-Lussac s Law 17

18 For constant temperature P 1 V Boyle s Law For constant pressure and temperature V N Avogadro s Law 18

19 Putting all of these statements together gives the ideal gas law (microscopic form): PV NkT k J/K is Boltzmann s constant The ideal gas law can also be written as (macroscopic form): PV nrt R N A k 8.31 J/K/mole is the universal gas constant and n is the number of moles. 19

20 Example (text problem 13.41): A cylinder in a car engine takes V i m 3 of air into the chamber at 30 C and at atmospheric pressure. The piston then compresses the air to one-ninth of the original volume and to 0.0 times the original pressure. What is the new temperature of the air? Here, V f V i /9, P f 0.0P i, and T i 30 C 303 K. P V i i P V f f NkT i NkT f The ideal gas law holds for each set of parameters (before compression and after compression). 0

21 1 Example continued: Take the ratio: i f i f i i f f T T NkT NkT PV P V The final temperature is ( ) K K i i i i i i f i f f V V P P T V V P P T The final temperature is 673 K 400 C.

22 13.6 Kinetic Theory of the Ideal Gas An ideal gas is a dilute gas where the particles act as point particles with no interactions except for elastic collisions.

23 Gas particles have random motions. Each time a particle collides with the walls of its container there is a force exerted on the wall. The force per unit area on the wall is equal to the pressure in the gas. The pressure will depend on: The number of gas particles Frequency of collisions with the walls Amount of momentum transferred during each collision 3

24 The pressure in the gas is P 3 N V K tr Where <K tr > is the average translational kinetic energy of the gas particles; it depends on the temperature of the gas. K tr 3 kt 4

25 The average kinetic energy also depends on the rms speed of the gas where the rms speed is 1 K tr m v 1 mv rms K tr v rms 3 kt 1 3kT m mv rms 5

26 The distribution of speeds in a gas is given by the Maxwell- Boltzmann Distribution. 6

27 Example (text problem 13.60): What is the temperature of an ideal gas whose molecules have an average translational kinetic energy of J? K tr T 3 K 3k tr kt 1550 K 7

28 Example (text problem 13.70): What are the rms speeds of helium atoms, and nitrogen, hydrogen, and oxygen molecules at 5 C? v rms 3kT m On the Kelvin scale T 5 C 98 K. Element Mass (kg) rms speed (m/s) He H N O

29 13.7 Temperature and Reaction Rates For a chemical reaction to proceed, the reactants must have a minimum amount of kinetic energy called activation energy (E a ). 9

30 If 3 E a >> kt then only molecules in the high speed tail of Maxwell- Boltzmann distribution can react. When this is the situation, the reaction rates are an exponential function of T. reaction rates e E a kt 30

31 Example (text problem 13.76): The reaction rate for the hydrolysis of benzoyl-l-arginine amide by trypsin at 10.0 C is times faster than at 5.0 C. Assuming that the reaction rate is exponential, what is the activation energy? r 1 r e e E a E a kt kt 1 where T C 83 K and T 5 C 78 K; and r r. The ratio of the reaction rates is r r 1 exp Ea kt 1 + Ea kt 31

32 Example continued: Solving for the activation energy gives: E a r 1 k ln r 1 1 T T1 ( J/K) ln( 1.878) 1 78 K 1 83 K J 3

33 13.8 Collisions Between Gas Molecules On average, a gas particle will be able to travel a distance Λ 1 ( N V ) πd / before colliding with another particle. This is the mean free path. The quantity πd is the cross-sectional area of the particle. 33

34 After a collision, the molecules involved will have their direction of travel changed. Successive collisions produce a random walk trajectory. 34

35 Substances will move from areas of high concentration to areas of lower concentration. This process is called diffusion. In a time t, the rms displacement in one direction is: xrms Dt D is the diffusion constant (see table 13.3). 35

36 Example (text problem 13.81): Estimate the time it takes a sucrose molecule to move 5.00 mm in one direction by diffusion in water. Assume there is no current in the water. xrms Dt Solve for t t x D ( 3 ) m ( m /s) rms s 36

37 Summary Definition of Temperature Temperature Scales (Celsius, Fahrenheit, Absolute) Thermal Expansion Origin of Pressure in a Gas Ideal Gas Law Exponential Reaction Rates Mean Free Path 37