Lecture 24Temperature and Heat

Phases and Phase Changes

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)

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.

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)

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.

- 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)

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)

(b)

ConvectionConvection 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.

RadiationThe amount of energy radiated by an object due to its temperature is proportional to its surface area and also to the fourth (!) power of its temperature.

It also depends on the emissivity, which is a number between 0 and 1 that indicates how effective a radiator the object is; a perfect radiator would have an emissivity of 1.

Here, e is the emissivity, and σ is the Stefan-Boltzmann constant:

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 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 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°).

Phases and Phase Changes

Ideal Gases

Gases are the easiest state of matter to describe, as all ideal gases exhibit similar behavior.

An ideal gas is one that is thin enough, and far away enough from condensing, that the interactions between molecules can be ignored.

Soda Bottle Soda Bottle

a) it expands and may burst

b) it does not change

c) it contracts and the sides collapse

inward

d) it is too dark in the fridge to tell

A plastic soda bottle is empty and sits out in the sun, heating the air inside. Now you put the cap on tightly and put the bottle in the fridge. What happens to the bottle as it cools?

The air inside the bottle is warmair inside the bottle is warm, due to heating by the

sun. When the bottle is in the fridge, the air coolsthe air cools. As

the temperature drops, the pressure in the bottle also pressure in the bottle also

dropsdrops. Eventually, the pressure inside is sufficiently

lower than the pressure outside (atmosphere) to begin

to collapse the bottle.

Soda Bottle Soda Bottle

a) it expands and may burst

b) it does not change

c) it contracts and the sides collapse

inward

d) it is too dark in the fridge to tell

A plastic soda bottle is empty and sits out in the sun, heating the air inside. Now you put the cap on tightly and put the bottle in the fridge. What happens to the bottle as it cools?

If the volume of an ideal gas is held constant, we find that the pressure increases with temperature:

Constant Volume

Constant Temperature

If the volume and temperature are kept constant, but more gas is added (such as in inflating a tire or basketball), the pressure will increase:

If the temperature is constant and the volume decreases, the pressure increases:

(fixed volume V) (fixed number N)

where k is called the Boltzmann constant:

Equation of State for an Ideal Gas

Combining these observations:

Instead of counting molecules, we can count moles.

Equation of State for an Ideal Gas

for n moles of gas:

Charles’s law says that the volume of a gas increases with temperature if the pressure is constant.

Properties of Ideal Gases: Constant Pressure

Boyle’s law says that the pressure varies inversely with volume. These curves of constant temperature are called isotherms.

Properties of Ideal Gases: Constant Temperature

Kinetic Theory

Assumptions:

• N identical molecules of mass m are inside a container of volume V; each acts as a point particle.

• Molecules always obey Newton’s laws and are moving randomly.

• Collisions with other molecules and with the walls are elastic.

The kinetic theory relates microscopic quantities (position, velocity) to macroscopic ones (pressure, temperature).

Containing an Ideal Gas

What is the impulse to turn a molecule around?

What is the period to time between such impulses?

I = 2(mvx)

t = 2L / vx

What is the average force on the surface from one molecule?

F = I/t = 2(mvx) / (2L/vx) = 2 mvx2 / L

Pressure and an Ideal Gas

Pressure is the result of collisions between the gas molecules and the walls of the container.

It depends on the mass and speed of the molecules, and on the container size:

Distribution of Molecular Speeds

Not all molecules in a gas will have the same speed; their speeds are represented by the Maxwell distribution, and depend on the temperature and mass of the molecules.

Pressure and Kinetic EnergyWe replace the speed in the previous expression for pressure with the average speed-squared:

Therefore, the pressure in a gas is proportional to the average kinetic energy of

its molecules.

Including the other two directions, and all N particles:

Kinetic Energy and TemperatureCompare to ideal gas law:

average kinetic energy is related to temperature

The square root of is called

the root mean square (rms) speed.

r.m.s. Speed

The rms speed is slightly greater than the most probable speed and the average speed.

Internal Energy

In the case where each molecule consists of a single atom, this is all linear kinetic energy of atoms:

The internal energy of an ideal monotonic gas is the sum of the kinetic energies of all its

molecules.

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

Solids

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; Y 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)

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

EvaporationMolecules in a liquid can sometimes

escape the binding forces and become vapor (gas)

Phase Equilibrium

If 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 PotatoesBoiling 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 PotatoesBoiling 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 Diagram

The 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 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.

You’re in Hot Water!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/1 cal/gg 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!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...

a) Yes

b) No

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

Boiling PotatoesBoiling Potatoes

Will potatoes cook faster if the water is boiling faster?

Boiling PotatoesBoiling 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?

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 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/gcwater = 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 m xm xTT = (1 cal/ = (1 cal/gg °°C) C) (1000 (1000 gg) ) (50 (50°°C) = 50,000 calC) = 50,000 cal

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

Water and 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/gcwater = 1 cal/g °C

a) 0°C

b) between 0°C and 50°C

c) 50°C

d) greater than 50°C

Follow-upFollow-up: : How much more water at 50°C would you need?