physics 1501: lecture 36, pg 1 physics 1501: lecture 36 today’s agenda l announcements çhomework...

Physics 1501: Lecture 36, Pg 1

Physics 1501: Lecture 36Physics 1501: Lecture 36TodayToday’’s Agendas Agenda

AnnouncementsHomework #12 (Dec. 9): 2 lowest droppedMidterm 2 … in class Wednesday

Honors’ students: Wednesdat at 3:30 in my office.

Today’s topicsChap. 17: ideal gas

» Kinetic theory

» Ideal gas law

» Diffusion Chap.18: Heat and Work

» Zeroth Law of thermodynamics» First Law of thermodynamics and applications» Work and heat engines


Kinetic Theory of an Ideal GasKinetic Theory of an Ideal Gas Pressure is


Does a Single Particle Have a Temperature?

Each particle in a gas has kinetic energy. On the previous page, we have established the relationship between the average kinetic energy per particle and the temperature of an ideal gas.

Is it valid, then, to conclude that a single particle has a temperature?

Concept of temperatureConcept of temperature


Air is primarily a mixture of nitrogen N2 molecules (molecular mass 28.0u) and oxygen O2 molecules (molecular mass 32.0u). Assume that each behaves as an ideal gas and determine

the rms speeds of the nitrogen and oxygen molecules when the temperature of the air is 293K.

For nitrogen

Example: Example: Speed of Molecules in Air


Internal energy of a monoatomic ideal gasInternal energy of a monoatomic ideal gas The kinetic energy per atom is

Total internal energy of the gas with N atoms


Kinetic Theory of an Ideal Gas: summaryKinetic Theory of an Ideal Gas: summary

Assumptions for ideal gas:Number of molecules N is largeThey obey Newton’s laws (but move

randomly as a whole)Short-range interactions during elastic

collisionsElastic collisions with wallsPure substance: identical molecules

Temperature is a direct measure of average kinetic energy of a molecule

Microscopic model for a gas Goal: relate T and P to motion of the

molecules


Theorem of equipartition of energy

Each degree of freedom contributes kBT/2 to the energy of a system (e.g., translation, rotation, or vibration)

Kinetic Theory of an Ideal Gas: summaryKinetic Theory of an Ideal Gas: summary

Total translational kinetic energy of a system of N molecules

Internal energy of monoatomic gas: U = Kideal = Ktot trans

Root-mean-square speed:


Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2.

11) If T is doubled, what happens to the rate at which ) If T is doubled, what happens to the rate at which a single a single moleculemolecule in the gas has a wall bounce? in the gas has a wall bounce?

b) x2a) x1.4 c) x4

22) If N is doubled, what happens to the rate at which ) If N is doubled, what happens to the rate at which a single a single moleculemolecule in the gas has a wall bounce? in the gas has a wall bounce?

b) x1.4a) x1 c) x2

Lecture 36: Lecture 36: ACT 1ACT 1


DiffusionDiffusion The process in which molecules move from a region

of higher concentration to one of lower concentration is called diffusion.Ink droplet in water


Why is diffusion a slow process ?Why is diffusion a slow process ? A gas molecule has a translational rms speed of hundreds of

meters per second at room temperature. At such speed, a molecule could travel across an ordinary room in just a fraction of a second. Yet, it often takes several seconds, and sometimes minutes, for the fragrance of a perfume to reach the other side of the room. Why does it take so long?Many collisions !


Comparing heat and molecule diffusionComparing heat and molecule diffusion Both ends are maintained at constant concentration/temperature


concentration gradientbetween ends

diffusion constant

SI Units for the Diffusion Constant: m2/s

FickFick’’s law of diffusions law of diffusion For heat conduction between two side at constant T

L

Th Tc A

Energy flow

conductivity temperature gradientbetween ends

The mass m of solute that diffuses in a time t through a solvent contained in a channel of length L and cross sectional area A is


Chap. 18: Work & 1Chap. 18: Work & 1stst Law Law

The Laws of Thermodynamics

0) If two objects are in thermal equilibrium with a third, they are in equilibrium with each other.

1) There is a quantity known as internal energy that in an isolated system always remains the same.

2) There is a quantity known as entropy that in a closed system always remains the same (reversible) or increases (irreversible).


Zeroth Law of ThermodynamicsZeroth Law of Thermodynamics

Thermal equilibrium: when objects in thermal contact cease heat transfersame temperature

T1T2

U1 U2

=

If objects A and B are separately in thermal equilibrium with a third object C, then objects A and B are in

thermal equilibrium with each other.

AC

B


11st st Law: Work & HeatLaw: Work & Heat Two types of variables

State variables: describe the system

(e.g. T, P, V, U). Transfer variables: describe the process

(e.g. Q, W).

=0 unless a process occurs

change in state variables. Work done on gas

W = F d cos = -F y

= - PA y = - P Vvalid only for isobaric processes

(P constant)If not, use average force or calculus:

W = area under PV curve

PV diagram


11st st Law: Work & HeatLaw: Work & Heat Work:

Depends on the path taken in the PV-diagramSame for Q (heat)


b) = |W1|a) > |W1| c) < |W1|

i

f

p

V

21

Consider the two paths, 1 and 2, connecting points i and f on the pV diagram. The magnitude of the work, |W2|,

done by the system in going from i to f along path 2 is

Lecture 36: Act 2Lecture 36: Act 2WorkWork


First Law of ThermodynamicsFirst Law of Thermodynamics

Isolated systemNo interaction with surroundingsQ = W = 0 U = 0.Uf = Ui : internal energy remains constant.

First Law of Thermodynamics

U = Q + W

variation of internal energyheat flow “in” (+) or “out” (-)

work done “on” the system

Independent of path in PV-diagramDepends only on state of the system (P,V,T, …)Energy conservation statement only U changes


Other ApplicationsOther Applications

Cyclic process:Process that starts and ends at the stateMust have U = 0 Q = -W .

Adiabatic process:No energy transferred through heat Q = 0.So, U = W .Important for

» expansion of gas in combustion engines

» Liquifaction of gases in cooling systems, etc. Isobaric process: (P is constant)

Work is simply


Other Applications (continued)Other Applications (continued) Isovolumetric process:

Constant volume W =0.So U = Q all heat is transferred into internal energy

» e.g. heating a “can” (no work done).

Isothermal process:T is constantUsing PV=nRT, we find P= nRT/V.Work becomes :

PV is constant.PV-diagram: isotherm.


p

V

Identify the nature of paths A, B, C, and D Isobaric, isothermal, isovolumetric, and adiabatic

Lecture 36: Act 3Lecture 36: Act 3ProcessesProcesses

A

B

C

D

T1

T2

T3T4


Heat EnginesHeat Engines We now try to do more than just raise the temperature of an

object by adding heat. We want to add heat to get some work done!

Heat engines: Purpose: Convert heat into work using a cyclic process Example: Cycle a piston of gas between hot and cold

reservoirs* (Stirling cycle)

1) hold volume fixed, raise temperature by adding heat

2) hold temperature fixed, do work by expansion

3) hold volume fixed, lower temperature by draining heat

4) hold temperature fixed, compress back to original V


Heat EnginesHeat Engines Example: the Stirling cycle

GasT=TH

GasT=TH

GasT=TC

GasT=TC

V

P

TC

TH

Va Vb

1 2

34

We can represent this cycle on a P-V diagram:

11 2

34*reservoir: large body whose temperature does not change when it absorbs or gives up heat


Identify whether Heat is ADDED or REMOVED from the

gas Work is done BY or ON the gas for each

step of the Stirling cycle:

V

P

TC

TH

Va Vb

1 2

34

ADDEDREMOVED

BYON

1

HEAT

WORK

step

ADDEDREMOVED

BYON

2

ADDEDREMOVED

BYON

3

ADDEDREMOVED

BYON

4

Heat EnginesHeat Engines

physics 1501: lecture 36, pg 1 physics 1501: lecture 36 today’s agenda l announcements çhomework...

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