Temperature and Heat. Heat and Mechanical Work Heat is another form of energy. James Joule used a...

Temperature and Heat

Heat and Mechanical Work

Heat is another form of energy.

James Joule used a device similar to this one to measure the mechanical equivalent of heat:

One kilocalorie (kcal) is defined as the amount of heat needed to raise the temperature of 1 kg of water from 14.5° C to 15.5° C.

Heat Capacity

The heat capacity of an object is the amount of heat added to it divided by its rise in temperature:

Q is positive if ΔT is positive; that is, if heat is added to a system.

Q is negative if ΔT is negative; that is, if heat is removed from a system.

Q = C ΔTHeat capacity tells you how much heat flow for a given ΔT

Specific Heat

The heat capacity of an object depends on its mass and on a property of the material itself: the specific heat

“heat capacity per kilogram”

Specific heats of various materials

C = mc

Heat capacity is mass x specific heat

A ceramic coffee cup, with c=1090 J/(kg K) and m =116 g, is initially at room temperature (24.0 °C). If 225 g of 80.3 °C coffee and 12.2 g of 5.00 °C cream are added to the cup, what is the equilibrium temperature of the system? Assume that no heat is exchanged with the surroundings, and that the specific heat of coffee and cream are the same as the specific heat of water. cwater = 4186 J / (kg K)

Thermal equilibrium is reached by means of thermal contact, which in turn can occur through three different mechanisms

Heat Transfer Mechanisms

conduction : it occurs when objects at different temperature are in physical contact (e.g. when holding a hot potato). Faster moving molecules in the hotter object transfer some of their energy to the colder one

convection : this occurs mainly in fluids. In a pot of water on a stove, the liquid at the bottom is heated by conduction. The hot water has lower density and rises to the top, cold water from the top falls to the bottom and gets heated, etc.

radiation : any object at non-zero temperature emits radiation (in the form of electromagnetic waves). The effect is more noticeable when standing next to a red-hot coal fire, or in the sun rays

ConductionConduction is the flow

of heat directly through a physical material

The amount of heat Q that flows through a rod:• increases proportionally to the cross-sectional area A• increases proportionally to ΔT from one end to the other• increases steadily with time• decreases inversely with the length of the rod

The constant k is called the thermal conductivity of the material

Some Typical Thermal Conductivities

Substances with high thermal conductivities are good conductors of heat; those with low thermal conductivities are good insulators.

Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is 0.525 m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C. (a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain. (b) find the temperature at the lead-copper interface.

kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)


Assumptions: •The end points are infinite heat reservoirs... so their temperature doesn’t change for this exercise •The temperature is constant in time at every point. This is not true at moment of thermal connection. We are solving the “steady state” condition, when the temperature at each point doesn’t change.

kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)


- The heat (per unit time) through the lead must equal that through the copper

- The lead has a smaller thermal conductivity than the copper

The lead requires a larger temperature difference across it than the copper, to get the same heat flow. So TJ > 54o C

kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)

(a)


kPb = 34.3 W / (kg-m)kCu = 395 W / (kg-m)

(b)

Convection

Convection is the flow of fluid due to a difference in temperatures, such as warm air rising. The fluid “carries” the heat with it as it moves.

Objects that are hot enough will glow – first red, then yellow, white, and blue.

RadiationAll objects give off energy in the form of radiation, as electromagnetic waves (light) – infrared, visible light, ultraviolet – which, unlike conduction and convection, can transport heat through a vacuum.

The surface of the Sun has a temperature of 5500 oC. (a) Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is 7.0x108 m, and the temperature of space can be taken to be 3.0 K (b) the solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is 1.5x1011 m.

The surface of the Sun has a temperature of 5500 oC. (a) Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is 7.0x108 m, and the temperature of space can be taken to be 3.0 K (b) the solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is 1.5x1011 m.

emissivity (a)

(b)

Heat Conduction

Given your experience of

what feels colder when you

walk on it, which of the

surfaces would have the

highest thermal

conductivity?

a) a rugb) a steel surfacec) a concrete floord) has nothing to do with

thermal conductivity

Heat Conduction

Given your experience of

what feels colder when you

walk on it, which of the

surfaces would have the

highest thermal

conductivity?

a) a rugb) a steel surfacec) a concrete floord) has nothing to do with

thermal conductivity

The heat flow rate is k A (T1 − T2)/l. All things being

equal, bigger k leads to bigger heat loss.From the book: Steel = 40, Concrete = 0.84,Human tissue = 0.2, Wool = 0.04, in units of J/(s.m.C°).

Chapter 17

Solids

Phase Changes

Thermal Processes

Solids

We’ve been dealing with fluids (gases or liquids), but this chapter describes some behaviors from the solid phase of matter as well.

(The topics are not so deep, but thy are on the MCAT...)

Solids and Elastic Deformation

Solids have definite shapes (unlike fluids), but they can be deformed. Pulling on opposite ends of a rod can cause it to stretch:

Stretching / Compression of a SolidThe amount of stretching will depend on the force; Υ is Young’s modulus and is a property of the material:

The stretch is proportional to the force, and also to the original length

The same formula works for stretching or compression (but sometimes with a different Young’s modulus)

Note: Larger modulus = smaller change (for same force)

Shear ForcesAnother type of deformation is called a shear deformation, where opposite sides of the object are pulled laterally in opposite directions.

The “lean” is proportional to the force, and also to the original height

Shear Modulus

S is the shear modulus.

Uniform Compression

Here, the proportionality constant, B, is called the bulk modulus.

Under uniform pressure, an object will shrink in volume

Stress and Strain

The applied force per unit area is called the stress, and the resulting deformation is the strain. They are proportional to each other until the stress becomes too large; permanent deformation will then occur.

Phase Changes

Kinetic Theory and Ideal Gases

The square root of (v2)av is called

the root mean square (rms)

speed.

Maxwell Distribution of molecular speed at a given temperature

PiVi = PfVf

Isotherm

= N Kav

Distribution of Molecular SpeedSome molecules will have speeds exceeding the planetary escape velocity!

Lighter molecules will have higher speeds (at the same temperature) and so will leave the planet more quickly.

This is why less massive planets have thin, or no, atmosphere...

and why earth has little H2 in the atmosphere, but Jupiter has plenty

EvaporationMolecules in a liquid can sometimes

escape the binding forces and become vapor (gas)

Phase EquilibriumIf a liquid is put into a sealed container so that there is a vacuum above it, some of the molecules in the liquid will vaporize. Once a sufficient number have done so, some will begin to condense back into the liquid. Equilibrium is reached when the numbers in each phase remain constant.

Vapor PressureThe pressure of the gas when it is in equilibrium with the liquid is called the equilibrium vapor pressure, and will depend on the temperature.

A liquid boils at the temperature at which its vapor pressure equals the external pressure.

a) Charlottesville

b) Denver (the “mile high” city)

c) the same in both places

d) I’ve never cooked in Denver, so I really don’t know

e) you can boil potatoes?

Boiling Potatoes

Will boiled potatoes cook faster in Charlottesville or in Denver?

a) Charlottesville

b) Denver (the “mile high” city)

c) the same in both places

d) I’ve never cooked in Denver, so I really don’t know

e) you can boil potatoes?

Boiling Potatoes

Will boiled potatoes cook faster in Charlottesville or in Denver?

The lower air pressure in Denver means that the water will boil at a lower temperature... and your potatoes will take longer to cook.

Phase DiagramThe vapor pressure curve is

only a part of the phase diagram.

When the liquid reaches the critical point, there is no longer a distinction between

liquid and gas; there is only a “fluid” phase.

There are similar curves describing the pressure/temperature of transition from solid to liquid, and solid to gas

Fusion CurveThe fusion curve is the boundary between the solid and liquid phases; along that curve they exist in equilibrium with each other.

One of these two fusion curves has a shape that is typical for most materials, but the other has shape specific to water.

Which is which?

(a) Curve 1 is the fusion curve for water

(b) Curve 2 is the fusion curve for water

(c) Trick question: there is no fusion curve for water!

Curve 1

Curve 2

Fusion curve for water

Ice melts under pressure!This is how an ice skate works

Phase EquilibriumThe sublimation curve marks the boundary between the solid and gas phases.

The triple point is where all three phases are in equilibrium.

Heat and Phase ChangeWhen two phases coexist, the temperature remains the same even if a small amount of heat is added. Instead of raising the temperature, the heat goes into changing the phase of the material – melting ice, for example.

Latent HeatThe heat required to convert from one phase to another is called the latent heat.

The latent heat, L, is the heat that must be added to or removed from one kilogram of a substance to convert it from one phase to another. During the conversion process, the temperature of the system remains constant.

Latent HeatThe latent heat of fusion is the heat needed to go from solid to liquid;

the latent heat of vaporization from liquid to gas.

Boiling Potatoes

Will potatoes cook faster if the water is boiling faster?

a) Yes

b) No

c) Wait, I’m confused. Am I still in Denver?

Boiling Potatoes

Will potatoes cook faster if the water is boiling faster?

The water boils at 100°C and remains at that temperature until allof the water has been changed into steam. Only then will the steam increase in temperature. Because the water stays at the same temperature, regardless of how fast it is boiling, thepotatoes will not cook any faster.

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

a) Yes

b) No

c) Wait, I’m confused. Am I still in Denver?

You’re in Hot Water!

Which will cause more

severe burns to your skin:

100°C water or 100°C

steam?

a) water

b) steam

c) both the same

d) it depends...

Although the water is indeed hot, it releases only 1 cal/(gK1 cal/(gK)) of

heat as it cools. The steam, however, first has to undergo a

phase changephase change into water and that process releases 540 cal/g540 cal/g,

which is a very large amount of heat. That immense release

of heat is what makes steam burns so dangerous.

You’re in Hot Water!

Which will cause more

severe burns to your skin:

100°C water or 100°C

steam?

a) water

b) steam

c) both the same

d) it depends...

Phase Changes and Energy Conservation

Solving problems involving phase changes is similar to solving problems involving heat transfer, except that the latent heat must be included as well.

Water and Ice

You put 1 kg of ice at 0°C

together with 1 kg of water

at 50°C. What is the final

temperature?

– LF = 80 cal/g

– cwater = 1 cal/g °C

a) 0°C

b) between 0°C and 50°C

c) 50°C

d) greater than 50°C

How much heat is needed to melt the ice?

QQ = = mLmLff = (1000 = (1000 gg) ) (80 cal/ (80 cal/gg) = 80,000 cal) = 80,000 cal

How much heat can the water deliver by cooling from 50°°C to 0°°C?

QQ = = ccwaterwater mmΔΔTT = (1 cal/ = (1 cal/gg °°C) C) (1000 (1000 gg) ) (50 (50°°C) = 50,000 C) = 50,000

calcal

Thus, there is not enough heat available to melt all the ice!!

Water and Ice

You put 1 kg of ice at 0°C

together with 1 kg of water

at 50°C. What is the final

temperature?

– LF = 80 cal/g

– cwater = 1 cal/g °C

a) 0°C

b) between 0°C and 50°C

c) 50°C

d) greater than 50°C

a) Add more ice to the icewater

b) add salt to the icewater

c) hold the icewater in an evacuated chamber (vacuum)

d) Jump in the car and drive to a nearby convenience store

Ice Cold Root BeerYou have neglected to chill root beer for your son’s 5th-birthday party. You submerge the cans in a bath of ice and water as you start dinner. How can you hurry the cooling process?

a) Add more ice to the icewater

b) add salt to the icewater

c) hold the icewater in an evacuated chamber (vacuum)

d) Jump in the car and drive to a nearby convenience store

Ice Cold Root BeerYou have neglected to chill root beer for your son’s 5th-birthday party. You submerge the cans in a bath of ice and water as you start dinner. How can you hurry the cooling process?

Not a), because ice water at 1 atm is zero degrees, no matter the proportion of water and ice

Not c), because ice is less dense than water so you will raise the melting point when you reduce the pressure. This will allow the water to get a little warmer than 0o

Not d), because you’ll forget your wallet and it will end up taking more time

b) because salt interferes with the formation of ice. This barrier to the solid phase lowers the fusion temperature, and so reduces the temperature of the ice water. (This is why you salt the sidewalk in winter.)


Fusion curve for most stuff

remember: water is weird: it melts under pressure, and freezes under vacuum, when near the fusion curve

Again: explaining why putting the ice/water under vacuum won’t help the root beer chill faster

As pressure goes lower, the ice/water mixture will ride the fusion curve from point 1 to point 2.This implies that temperature goes up.


1

2ΔP

ΔT

The larger ΔT, the more heat transfers per unit time. Thus, the colder the ice bath, the faster the root beer will chill, and the warmer the bath, the slower the root beer will chill

When two states exist in the same system (like, ice and water), the system MUST be on the equilibrium curve (in the case, the fusion curve).

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