Physics 201 - University of Virginia

Transcript Physics 201 - University of Virginia

PHYS 1110

Lecture 9 Professor Stephen Thornton September 25, 2012

Reading Quiz

Two identical cylinders at the same temperature contain the same gas. If A contains three times as much gas as B , which cylinder has the higher pressure ?

A) cylinder A B) cylinder B C) both the same D) it depends on temperature T

Reading Quiz Two identical cylinders at the same temperature contain the same gas. If A contains three times as much gas as B , which cylinder has the higher pressure ?

A) cylinder A B) cylinder B C) both the same D) it depends on temperature T Ideal gas law: PV Solve for pressure: = nRT P = nRT V For constant V and T, the one with more gas ( the larger value of n ) has the higher pressure P.

Midterm exam next Tuesday.

Chapters 1-4.

Thermodynamics will not be on the exam.

Homework, including the electric motor, is due this Thursday.

Unit of inductance: the henry, H: 1 H = 1 V·s/A = 1 Ω·s.

A transformer is an example of mutual inductance.

Area

Solenoid Self-Induction

  

 0

nI NBA

  0

nNIA

  0 2

An I

 

B LI

, so

  0

Only depends on geometry.

Consider an inductor

For a

solenoid

U L

  0 2

n A

 1 2

2  1 2   0 2

n A



2  1 2  0 (  0 2 2

n I

2 )

But

  0

, so

U B

 1 2  0 2

B A

 1 2  0 2

B V u B

 magnetic energy volume 

2 2  0 energy density

General energy density

u B

 1 2

2  0 general result

u E

 1 2  0



u B



u E

 1 2  

2  0   0

2  

c

1

m e

0 0

We can produce an emf by using AC voltage and coils.

Show demo of AC coils and light bulb

Do transformer demo

Transformer equation Primary coil: 

Secondary coil:   

S N P

  

N S P







For good transformer,    P

and

 

P S



N P N S

If resistance is small,

V S



V P

 

N S N P V P



N P V S N S

  step-up and step-down transformers

This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary:

Lots of applications for transformers, the bug zapper.

Power distribution

Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac.

What is the voltage across the lightbulb?

Conceptual Quiz

A) 30 V B) 60 V C) 120 V D) 240 V E) 480 V 120 V

Conceptual Quiz

A) 30 V B) 60 V What is the voltage across the lightbulb? C) 120 V D) 240 V E) 480 V The first transformer has a 2:1 ratio of turns, so the voltage doubles . But the second transformer has a 1:2 ratio , so the voltage is halved again. Therefore, the end result is the same as the original voltage .

120 V 240 V 120 V

Conceptual Quiz

A 6 V battery is connected to one side of a transformer. Compared to the voltage drop across coil A, the voltage across coil B is: A) greater than 6 V B) 6 V C) less than 6 V D) zero

6 V

A B

Conceptual Quiz

A 6 V battery is connected to one side of a transformer. Compared to the voltage drop across coil A, the voltage across coil B is: A) greater than 6 V B) 6 V C) less than 6 V D) zero The voltage across B is zero . Only a changing magnetic flux induces an emf. Batteries can provide only dc current .

A B

6 V

thermal system

is a physical system having temperature-dependent properties. A piston and cylinder full of hot steam is a thermal system.

If we pour hot water into a bucket of cold water, the temperature eventually is the same throughout the system. We say they have reached

thermal equilibrium.

The system is then at a single, constant temperature.

thermal reservoir

(or heat bath) is a thermal system so large that it keeps a constant temperature when it interacts with other systems. A large lake is a thermal reservoir. The outside atmosphere is a thermal reservoir. Your bathtub full of water is not a thermal reservoir.

When systems come into

thermal contact

with each other, they can exchange energy.

Temperature, heat

concepts

  Temperature is not the same thing as heat.

Temperature refers to how “hot” or “cold” something is.

 Temperature is related to the internal energy or thermal energy of a system.

  Heat refers to a flow of energy. There is no such concept as absolute heat. Heat flow is more appropriate.

Thermometers It is not so easy to build good thermometers.

Criteria:  Must be small enough to not affect system being measured.

  Must have good thermal contact.

Reproducible, easily read, safe, widely available.

 Accurate over wide range of temperatures .

Comparison of Temperature Scales

Kelvin temperature Because temperature is a measure of the energy of a thermal system, it makes sense to have zero (0) as the lowest possible temperature.

1 C 0 temperature change = 1 K change At 0 K, all motion stops.

Conceptual Quiz

A) yes, at 0 °C It turns out that –40°C is the same temperature as –40°F. Is B) yes, at − 273 °C C) yes, at 0 K there a temperature at which the Kelvin and Celsius scales agree? D) no

Conceptual Quiz

A) yes, at 0 °C It turns out that –40°C is the same temperature as –40°F. Is B) yes, at − 273 °C C) yes, at 0 K there a temperature at which the Kelvin and Celsius scales agree? D) no The Celsius and Kelvin scales differ only by an offset, which is 273 degrees. Therefore, a temperature on one scale can never match the same numerical value on the other scale. The reason that such agreement is possible for Celsius and Fahrenheit is the fact that the actual degree units have different sizes (recall the previous question).

Most substances expand when heated.

We have all seen expansion joints in concrete sidewalks, on highways, on steel bridges, etc.

Consider a rod of length at 0 temperature

0 . When heated it expands (where  = coefficient of linear expansion [unit 0 C -1 )]  0 (1   

)

  

Do demos of linear expansion.

Start bolt cracker Do bimetallic strip – see next slide Ring and ball – do later

A Bimetallic Strip Do bimetallic strip demo.

Here metal B has a larger coefficient of linear expansion.

Now consider a plate of sides

Each side expands by

 

'  (

 

) 2  (

  

2  2  2    2

' 

2  2  2

L T

2  2  2 ) 2 0 2  area expansion If we do this for volume, say a cube 3   volume expansion  is the coefficient of volume expansion.

Conceptual Quiz: We sometimes pour hot water on a metal lid that seems to be stuck on a glass jar. Why do we do this?

A) Because glass expands more than the lid, which will loosen the lid.

B) Because the junk between the lid and the glass will melt and flow out.

C) Because the metal lid expands more than the glass, which will loosen the lid.

D) Because I once saw it done on television, and it worked.

Answer: C The thermal expansion coefficients of metal are much greater than that of glass. The hot water causes the metal lid to expand much more than the glass jar, which will loosen the lid.

Do ball through ring demo. Heat ring and see if ball goes through.

Heat capacity

We find that every substance has a specific property of heat capacity





0 unit: J/K or J/ C or

Q Q

is positive or negative depending on whether heat energy is added or removed from the object.

Specific Heat

• Heat capacity is not so useful because we must specify type and how much of substance.

• We define a new quantity, called

specific heat

that depends on the particular substance.



 0 C) For water:

water 

  heat required to raise temperature 

Conceptual Quiz: Pyrex glass is often used for baking dishes and even to boil water on a stove. The coefficient of linear expansion is much smaller for Pyrex than for ordinary glass. What is likely to happen if we put a glass jar and a Pyrex jar on a hot stove element?

A) The glass jar would break before the Pyrex.

B) The Pyrex would break before the glass jar.

C) They would break at about the same time if they are the same thickness.

D) If they have water in them, the Pyrex would break first.

Answer: A Because the glass jar has a higher coefficient of linear expansion the part of the glass that gets hot first would start expanding first, and this would put a good deal of stress on the glass structure. Because Pyrex has a lower value, it would not expand as much so quickly.

Latent heat L is the energy/kg needed to change the phase state of matter.

Q mL Latent heat of fusion L

F is energy/kg needed to melt a substance; solid goes to liquid.

Latent heat of vaporization L

V is energy/kg needed to go from liquid to gas.

Latent heat of sublimation L

S is energy/kg required to go from solid directly to gas.

Latent Heat

Energy is required for a material to change phase, even though its temperature is not changing.

1 kg

v = 539 cal/g = 539 kcal/kg

Latent Heat

The latent heat of vaporization is relevant for evaporation as well as boiling. On a molecular level, the heat added during a change of state does not go to increasing the kinetic energy of individual molecules, but rather to breaking the close bonds between them so the next phase can occur.

Conceptual Quiz

If you add some heat to a substance, is it possible for the temperature of the substance to remain unchanged?

A) yes B) no C) depends on Q D) depends on W

Conceptual Quiz

If you add some heat to a substance, is it possible for the temperature of the substance to remain unchanged?

A) yes B) no C) depends on Q D) depends on W Yes, it is indeed possible for the temperature to stay the same. This is precisely what occurs during a phase change – the added heat goes into changing the state of the substance (from solid to liquid or from liquid to gas) and does not go into changing the temperature! Once the phase change has been accomplished, then the temperature of the substance will rise with more added heat.

Follow-up: Does that depend on the substance?

Conceptual Quiz

Will potatoes cook faster if the water is boiling faster?

A) yes B) no C) depends on W

Conceptual Quiz

Will potatoes cook faster if the water is boiling faster?

A) yes B) no C) depends on W The water boils at 100 °C and remains at that temperature until all of the water has been changed into steam. Only then will the steam increase in temperature. Since the water stays at the same temperature, regardless of how fast it is boiling, the potatoes will not cook any faster.

Follow-up: How can you cook the potatoes faster?

Conceptual Quiz Water

has a higher specific heat than

sand

. Therefore, on the beach at night, breezes would blow: A) from the ocean to the beach B) from the beach to the ocean C) either way, makes no difference

Conceptual Quiz Water

has a higher specific heat than

sand

. Therefore, on the beach at night, breezes would blow: A) from the ocean to the beach B) from the beach to the ocean C) either way, makes no difference



Daytime



sun heats both the beach and the water

» » » »

beach heats up faster warmer air above beach rises cooler air from ocean moves in underneath breeze blows ocean



land



sand < c water Nighttime



sun has gone to sleep

» »

beach cools down faster warmer air is now above the ocean

» »

cooler air from beach moves out to the ocean breeze blows land



ocean

Why do we almost always feel a breeze at the beach?

See if this works this summer!

Water has high thermal conductivity!

Ideal gases

We will mostly discuss ideal gases, because it is easy to experiment with them. Air at atmospheric pressure is close to ideal. Intermolecular forces are negligible for ideal gases.

The relationship between the volume, pressure, temperature, and mass of a gas is called an equation of state .

V, P, T, m

are thermodynamic state variables.



NkT

Ideal gas law

where

is the Boltzmann constant and is 1.38 x 10 -23 J/K.

The temperature

must be kelvin K.

Sometimes pressure is lower case

A mole is the number of grams of a substance equal to the molecular weight of the substance.

For example, 1 mol of helium gas has mass 4 g 1 mol of oxygen gas has 32 g 1 mol of CO 2 has mass 44 g (We should use the more accurate molecular weights) One mole of gas always contains precisely

A = 6.022 x 10 23 molecules/mole ( Avogadro’s number) The number of moles in a certain mass of material is

(mole) = mass (grams) molecular mass (g/mol)

Let

= # of moles, then

N = nN

A = number of molecules

PV = NkT

becomes

PV = nN

We now define the universal gas constant

to be

R = N

k =

8.31 J/(mol K) so now,

Plot

= constant. Curves are called

isotherms

, because

is constant.

(constant pressure curves are called

isobars

.) isotherm

Conceptual Quiz

Two identical cylinders at the same pressure contain the same gas. If A contains three times as much gas as B , which cylinder has the higher temperature ?

A) cylinder A B) cylinder B C) both the same D) it depends on the pressure P

Conceptual Quiz

Two identical cylinders at the same pressure contain the same gas. If A contains three times as much gas as B , which cylinder has the higher temperature ?

A) cylinder A B) cylinder B C) both the same D) it depends on the pressure P Ideal gas law: PV = nRT Solve for temperature: T = PV nR For constant V and P, the one with less gas ( the smaller value of n ) has the higher temperature T.

Kinetic Theory of Gases

• James Clerk Maxwell, a great Scottish mathematical physicist, did much of the early work in statistical theory, but died at age 48.

• Ludwig Boltzmann despaired so much about his work in statistical theory that he committed suicide in 1906.

• Paul Ehrenfest, who picked up Boltzmann’s work, committed suicide in 1933.

• Now it is our turn to study the subject.

• Perhaps we should proceed cautiously!

Kinetic theory of gases

Assumptions: • Container has large number

of identical molecules, each of mass

• Molecules bounce around and are far apart.

• Molecules collide elastically with wall and each other.

Maxwell did a complete calculation using the average velocity and included molecules moving in all 3 directions.

He found

PV

 2 3

N

1 2

mv



 2 3

NK

The pressure is proportional to the average kinetic energy.

There is a clear connection between microscopic behavior and macroscopic variables.



NkT

 2 3

1 2

2 av So

kT mv

2 av 1 2

2    av

av  3 2

This indicates that the average kinetic energy

Speed Distribution for O 2 and H 2 at 20 ºC 1 2

2    av

av  3 2

Much easier for H molecules 2 to leave Earth than O 2

The molecules having higher speed are able to leave a sweat drop on the skin, resulting in a lower temperature and drawing heat from the skin. This in turn heats the drop and the process occurs again.

Heat and mechanical work

At one time it was thought that all substances contained a heat fluid or “caloric” that could flow from one substance to another.

The American royalist Benjamin Thompson (Count Rumford) finally figured out while boring cannons that friction caused heat flow, which is a form of energy.

Doing work causes heat flow, so they should be related.

Heat as Energy Transfer

We often speak of heat as though it were a material that flows from one object to another; it is not. Rather, it is a form of energy flow.

Unit of heat: calorie (cal) 1 cal is the amount of heat necessary to raise the temperature of 1 g of water by 1 Celsius degree.

Don’t be fooled—the calories on our food labels are really kilocalories (kcal or Calories), the heat necessary to raise 1 kg of water by 1 Celsius degree.

Heat as Energy Transfer

Definition of heat:

Heat is energy transferred from one object to another because of a difference in temperature.

Remember that the temperature of a gas is a measure of the kinetic energy of its molecules.

Mechanical equivalent of heat British physicist James Joule did experiment in mid-1800s to determine how heat flow is related to work and energy.

Now we know 1 cal = 4.186 J 1 Btu = 0.252 kcal = 1055 J

= heat flow = energy transferred due to temperature differences.

The Mechanical Equivalent of Heat

Internal Energy

The sum total of all the energy of all the molecules in a substance is its internal (or thermal) energy.

Temperature : measures molecules’ average kinetic energy Internal energy : total energy of all molecules Heat : transfer of energy due to difference in temperature Copyright © 2009 Pearson Education, Inc.

Internal energy of an ideal (monoatomic) gas (that is, its translational KE):

int =

ж 1 зи 2

2 цч ш

But since we know the average kinetic energy in terms of the temperature, we can write:

int = 3 2

NkT

We often denote internal energy by

.

Internal Energy

If the gas is molecular rather than atomic, rotational and vibrational kinetic energy need to be taken into account as well.

Heat Exchange

   Conduction – molecules touch each other and exchange energy.

Convection – hot fluids rise Radiation – electromagnetic radiation like light, infrared, ultraviolet radiation; all frequencies.

These are very important!!

Heat conduction

If we put a torch to a piece of metal, the molecules in the metal have increased kinetic energy. They collide with adjacent molecules, and the heat moves down the material via these collisions. Some materials transport heat energy more easily than others. Metals are good heat conductors. Wood and plastics are poor.

Heat Conduction Through a Rod

is heat flow through rod

Q is proportional to

and temperatures

2 Q is proportional to 1/

–





T L t k

is called the thermal conductivity W/(m K)

The constant

is called the thermal conductivity.

Materials with large

are called conductors; those with small

are called insulators.

Note: materials that are good heat conductors are also good electrical conductors. Why?

Note in the table on thermal conductivities that air is a very poor heat conductor. In fact, we could say it is a good heat insulator.

This is why double pane windows are such good insulators both in the summer and winter. Glass panes are thin and conduct heat much better than air. The layer of air does wonders!

Building materials are measured using R values rather than thermal conductivity:

= /

Here, is the thickness of the material.

Heat convection

Well known phenomenon because hot fluids rise due to their lower density. We take advantage of this by putting heat ducts on the floor. Do demo – convection chimney

Convection occurs when heat flows by the mass movement of molecules from one place to another. It may be natural or forced; both these examples are natural convection.

Heat radiation

Have you ever sat in front of a campfire and wondered why your face is so warm, and your behind so cold?

All objects emit electromagnetic radiation. Waves easily carry energy in the form of light, radar, microwave (cell phone), etc.

Our existence depends on heat radiation from the Sun.

Do light the match (wood) demo. Example of radiation.

Heat radiation is noted in terms of radiated power

P P

 



 4 unit W

is called the emissivity and is   -8 2  4 K ) Called the Stefan-Boltzmann constant

= 1 is a perfect emitter and absorber, and is called a blackbody.

= 0 is an ideal reflector.

Inside of a thermos bottle is shiny and is a good reflector. The heat of the container emits radiation, but it is not absorbed by the outer wall.

The Thermos Bottle

If you are in the sunlight, the Sun’s radiation will warm you. In general, you will not be perfectly perpendicular to the Sun’s rays, and will absorb energy at the rate: D

= D

1000

e A

cos

This cos θ effect is also responsible for the seasons.

Consider a system:

Automobile engine Human body Simple piston and cylinder We want to consider what happens if we add heat to our system or take heat away.

Also let the system do work or have work done on it.

What happens to internal energy? The internal energy is the sum of all the kinetic and potential energies.

The Internal Energy of a System

f =

i +

If we add heat

to a system having internal energy

i , the new internal energy of the system is

f =

i +

. 

f –

i =

Work and Internal Energy

f = E i -W

If the system does work

on the outside, then the system loses internal energy.

i –

f = W 

f –

i = -