Physics 2 week 7. Chapter 3 The Kinetic Theory of Gases

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1 Physics week 7 Chapter 3 The Kinetic Theory of Gases 3.1. Ideal Gases Experimental Laws and the Equation of State Molecular Model of an Ideal Gas 3.. Mean Free Path 3.3. The Boltzmann Distribution Law and The Distribution of Molecular Speeds 3.4. The Molar Specific Heats of an Ideal Gas 3.5. The Equipartition-of-Energy Theorem 3.6. The Adiabatic Expansion of an Ideal Gas

2 Reiew Number of moles in a sample: n n M sample M pv N N A Ideal gas law: or pv NkT M nrt sample mn A N is the number of molecules M is the molar mass (the mass of 1 mol) m is the mass of one molecule p is the absolute pressure (Pa) T is the temperature (in K) Gas constant (R) R 8.31 J mol 1 1 K The Boltzmann constant (k) k R N A J K -1

3 Reiew Work done by the gas: V W pdv Vi : 1) If V constant (isochoric) W 0 ) If p constant (isobaric) : W p(v V ) pv 3) If T constant (isotherma l) : f W nrt f i ln V f V i

4 13. A sample of an ideal gas is taken through the cyclic process abca shown in the figure below; at point a, T=00 K. (a) How many moles of gas are in the sample? (b) the temperature of the gas at point b and point c, (c) the net energy added to the gas as heat during the cycle? (a) Applying the equation of state: pv nrt n pv RT At point a, p=.5 kn/m or 500 N/m ; V=1 m n 1.5 (mol) pava pbvb nrt nr 1. T T (b) pv 5 a b

5 Point b, p=7.5 kn/m ; V=3 m 3 T b pbv nr (K) 1.5 Point c, p=.5 kn/m ; V=3 m 3 b T c pcv nr c (K) (c) Applying the first law of thermodynamics: E Q W W: work done by the system. For a closed cycle, E=0: W W 1 ( p b p c )( V b V 1 3 a ) 510 Q (J) W

6 3.1.. Molecular Model for an Ideal Gas In this model: 1. The molecules obey Newton's laws of motion.. The molecules moe in all direction with equal probability. 3. There is no interactions between molecules (no collisions between molecules). 4. The molecules undergo elastic collisions with the walls.

a. Pressure, Temperature, and RMS Speed Problem: Let n moles of an ideal gas be confined in a cubical box of olume V. The walls of the box are held at temperature T.

7 a. Pressure, Temperature, and RMS Speed Problem: Let n moles of an ideal gas be confined in a cubical box of olume V. The walls of the box are held at temperature T. Key question: What is the connection between the pressure p exerted by the gas and the speed of the molecules? First, we consider a cubical box of edge length L, containing n moles of an ideal gas. A molecule of mass m and elocity collide with the shaded wall. For an elastic collision, the particle s momentum (=m.) along the x axis is resered and change with an amount: Δp x ( m x ) (m x ) m x

8 The aerage rate at which momentum is deliered to the shaded wall by this molecule: Trael time b/w walls with a speed Recall: F ma d m dt Δp Δt x m L/ d(m) dt x x dp dt m L x F x,1 The pressure exerted on the wall by this single molecule: m L x p 1 F x,1 L Note: Pressure is the force applied perpendicular to the surface of an object For N molecules, the total pressure p: p F L x m x,1 /L m x, /L L... m x, N /L

9 p m (x,1 x,... 3 L The aerage alue of the square of the x components of all the molecular speeds: nmn p 3 L M mn Since V L For any molecule: 3 : A x A x :the p x,1 x,... N molar nm V mass x of z the As all molecules moe in random directions: x y x, N gas x, N ) x 1 3

10 The square root of p nm 3V is called the root-mean-square speed: p rms nm 3V rms This relationship shows us how the pressure of the gas (a macroscopic quantity) depends on the speed of the molecules (a microscopic quantity) Combining with the equation of state: pv nrt rms 3RT M

11 b. Translational Kinetic Energy Consider a single molecule of an ideal gas moing around in the box (see Section a). K K 1 1 m K m 1 3RT M 3RT N A m 1 The Boltzmann constant k: x K y 3 kt z RT M/m m k rms R N A Kdoes not depend on the mass of the molecule 1 m m m x y z kt

12 4. At 73 K and 1.0 x 10 - atm, the density of a gas is 1.4 x 10-5 g/cm 3. (a) Find rms for the gas molecules. (b) Find the molar mass of the gas and (c) identify the gas (hint: see Table 19-1). (a) Root-mean-square speed: (1) and (): M gas rms 3RT M nm M V V 3nRT 3 rms V (1) V n p () g/cm kg/m 3 p atm Pa rms 494 m/s

13 (b) Equation of state: (c) M V n pv nrt M V n () (3) M 0.08 kg/mol RT p 8 g/mol From Table 19.1, the gas is nitrogen (N )

14 Homework: 18, 0, 3, 5, 7 (p )

15 Physics week 8 Chapter 3 The Kinetic Theory of Gases 3.1. Ideal Gases Experimental Laws and the Equation of State Molecular Model of an Ideal Gas 3.. Mean Free Path 3.3. The Boltzmann Distribution Law and The Distribution of Molecular Speeds 3.4. The Molar Specific Heats of an Ideal Gas 3.5. The Equipartition-of-Energy Theorem 3.6. The Adiabatic Expansion of an Ideal Gas

3.. Mean Free Path 3..1 Concept A molecule traeling through a gas changes both speed and direction as it elastically collides with other molecules in its path.

16 3.. Mean Free Path 3..1 Concept A molecule traeling through a gas changes both speed and direction as it elastically collides with other molecules in its path. Between collisions, the molecules moes in a straight line at constant speed. The mean free path is the aerage distance traersed by a molecule between collisions. λ 1 density 1 N V where V is the olume of N is the number of the gas molecules N is the number of molecules per unit olume or the V density of molecules

+ All molecules are spheres of diameter d a collision occurs as the centers of molecules come within a distance d.

17 Our goal: Estimate of of a single molecule. Assumptions: + Our molecule is traeling with a constant speed and all the other molecules are at rest. + All molecules are spheres of diameter d a collision occurs as the centers of molecules come within a distance d. To count the number of collisions: We further consider that this single molecule has an equialent radius of d and all the other molecules are points (see cartoons next slides for an equialent problem).

18 Equialent problem = 1 collision d. = 1 collision d

19 3 rd collision 1 st collision d nd collision Equialent problem d

20 The number of collisions = the number molecules lie in a cylinder of length t and cross-sectional area d

21 If all the molecules are moing: 1 d N V Using the equation of state: pv = NkT kt d p The aerage time between collisions (the mean free time): kt d t p The frequency of collisions: f 1 t

22 3.3. The Boltzmann Distribution Law and the Distribution of Molecular Speeds The Boltzmann distribution law: if the energy is associated with some state or condition of a system is then the frequency with which that state or condition occurs, or the probability of its occurrence is proportional to: e / kt k : the Boltzmann constant Many of the most familiar laws of physical chemistry are special cases of the Boltzmann distribution law:

23 The distribution of molecular speeds (or the Maxwell speed distribution law): Let M be the molar mass of the gas, be the molecular speed, and P() be the speed distribution function: P( ) 4 M RT 3/ P()d is the fraction of molecules with speeds in the infinitesimal range (,+d). e P( ) d 1 0 The fraction of molecules with speeds from 1 to : frac M / kt 1 (1) P( ) d

24 Aerage, RMS, and Most Probable Speeds The aerage speed: from (1) & (): 0 RT 3 M P( ) d 8RT M 0 P( ) d The root-mean-square speed: () 3RT rms M The most probable speed is the speed at which P() is maximum: dp( ) d 0 RT P M

25 3.3.. The barometric distribution law: This law gies the number density (h), i.e. number of molecules per unit olume, of an ideal gas of uniform temperature T as a function of height h in the field of the Earth s graity. ( h) ( h0) e mg ( hh0 ) / kt where h 0 is an arbitrary fixed reference height; m is the mass of a molecule. nasa.go Abell 198

26 Homework: 8, 3, 33, 40 (Page )

27 Physics week 9 Chapter 3 The Kinetic Theory of Gases 3.1. Ideal Gases Experimental Laws and the Equation of State Molecular Model of an Ideal Gas 3.. Mean Free Path 3.3. The Boltzmann Distribution Law and The Distribution of Molecular Speeds 3.4. The Molar Specific Heats of an Ideal Gas 3.5. The Equipartition-of-Energy Theorem 3.6. The Adiabatic Expansion of an Ideal Gas

28 3.4. The Molar Specific Heats of an Ideal Gas Let s consider our ideal gas of n moles that is a monatomic gas, which has indiidual atoms, e.g. helium, argon, neon. For a single atom, the aerage translational KE: 3 K kt The internal energy E int of the gas (no rotational KE for monatomic gases): E int N 1 K 3 kt Recall molar specific heat: nn a. Molar specific heat at constant olume: 3 nrt Þ DE int = 3 nrdt Q CnT A Consider n moles of an ideal gas at state i: p, T, and fixed V state f: p+p, T+T Q nc V C V is a constant and called the molar specific heat at constant olume. T

29 3 Eint Q W ncv T W nrt 3 Since W 0 CV R 1.5 J mol Note: For diatomic and polyatomic gases, their C V is greater than that of monatomic gases. So, the change in internal energy can be calculated by: or E int E int 3 nc nrt V T 1 K 1

30 b. Molar specific heat at constant pressure: Q E int nc pt C p is the molar specific heat at constant pressure. 3 W pv Q W nrt nrt nc pt nrt C 3 5 R R p R C p C V R

31 Checkpoint 4 (p. 5): The figure here shows 5 paths traersed by a gas on a p-v diagram. Rank the paths according to the change in internal energy of the gas, greatest first. E int 3 nrt T T T 3 1

32 Example: (Problem 8, page 530) Suppose 1.8 mol of an ideal gas is taken from a olume of 3.0 m 3 to a olume of 1.5 m 3 ia an isothermal compression at 30 0 C. (a) How much energy is transferred as heat during the compression, and (b) is the transfer to or from the gas? (a) We hae: An isothermal process: T=constant Work done by the gas for isotherm: (b) Q<0: heat transferred from the gas

33 3.5. The Equipartition-of-Energy Theorem Eery kind of molecule has a certain number f of degrees of freedom. For each degree of freedom in which a molecule can store energy, the aerage internal energy is 1 kt per molecule. Degrees of freedom Molecule Monatomic Example He Translational Rotational Total (f) Diatomic O Polyatomic CH C C V p f C V R R

34 Six degrees of freedom Technical Aspects of robotics wac.nsw.edu.au

35 5 degrees of freedom of a diatomic molecule

36 3.6. The Adiabatic Expansion of an Ideal Gas What is an adiabatic process?: a process for which Q = 0 pv constant where C p /C V Equation of state: pv nrt TV 1 constant Proof of the equations aboe, see p (homework) Free expansions: Recall : Q W 0 E 0 int p V i i T i T f p f V f

37 Homework: 4, 44, 46, 54, 56, 78 (p )

Homework: 13, 14, 18, 20, 24 (p )

Homework: 13, 14, 18, 20, 24 (p ) Homework: 13, 14, 18, 0, 4 (p. 531-53) 13. A sample of an ideal gas is taken through the cyclic process abca shown in the figure below; at point a, T=00 K. (a) How many moles of gas are in the sample?