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Chapter 17

Temperature and Heat

17.1 Temperature and Thermal Equilibrium

Temperature is a measure of how hot or cold something

is. Two systems are said to be in thermal equilibrium with

one another if and only if they have the same temperature.

The Zeroth Law of Thermodynamics

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 another.

17.2 and 17.3 Thermometers and Temperature Scales

Thermometers are used to measure temperature by the

expansion and contraction of a liquid (such as mercury).

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32TT CF 59 +=

( )32TT FC 9

5!=

15.273TT C +=

TF = Temperature in Fahrenheit TC = Temperature in Celsius T = “Absolute temperature” in Kelvin Absolute zero !state of minimum but finite atomic motion. Zero-point energy !energy associated with the motion of atoms at 0 kelvin.

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17.4 Thermal Expansion of Solids and Liquids

A. Linear Expansion of Solids The length

!

Loof an object changes by an amount L! when

its temperature changes by an amount T! :

!

"L =# L0 "T (solids)

where !" thermal coefficient of linear expansion of the

object.

For many materials, every linear dimension changes

according to the formula above. Thus L could be the

thickness of a rod, the side length of a square sheet, or the

diameter of a hole.

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B. Volume Expansion of liquids and solids

The volume

!

Voof an object changes by an amount

V! when its temperature changes by an amount T! :

!

"V = #V0 "T (liquids and solids)

where !" thermal coefficient of volume expansion of the object. • Expansion slots on bridges are needed to

accommodate changes in length that result from

thermal expansion.

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• Running hot water on a jar lid loosens it. This is

because the metal may be struck more directly by the

hot water, but even if not, metals expand more than

glasses because glassmetal !>!

• A hole in a piece of material expands when heated

and contracts when cooled, just as if it were filled

with the material that surrounds it.

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Tables 17.1 and 17.2 list the values of the coefficients of

expansion of various materials. Note that for many solids,

!

" # 3$ .

Thermal Expansion of Water:

The anomalous behavior of water between 0oC and 4oC:

Water actually contracts as its temperature increases from

0 oC to 4 oC, and thus its density increases in this

temperature range. It is this weird behavior of water in

this temperature range that explains why lakes freeze

from the top down.

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Notice that above 4 oC, water exhibits the “expected”

expansion with increasing temperature.

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Thermal Stress

If we clamp the ends of a rod rigidly to prevent expansion

or contraction and then change the temperature, thermal

stresses develop.

To calculate the thermal stress in a clamped rod, we

compute the amount the rod would expand (or contract) if

not clamped and then calculate the stress needed to

compress (or stretch) it back to its original length.

Suppose that a rod of length Lo and cross-sectional area A

is held at constant length while the temperature is reduced

by ΔT, causing a tensile stress. Since the length is to be

constant, then the total fractional change in length must

be zero.

!

"LL0

#

$ %

&

' ( thermal

+"LL0

#

$ %

&

' ( tension

= 0

!

"#T +FAY

= 0

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!

FA

= "Y #$T Thermal Stress

where Y = Young’s modulus of the material.

Note that for a decrease in temperature, ΔT is negative, so

F and F/A are positive, meaning that a tensile force and

stress are needed to maintain the length. On the other

hand, for an increase in temperature, ΔT is positive, so F

and F/A are negative, meaning that a compressive force

and stress are needed to maintain the length.

17.5 Quantity of Heat

Heat is the thermal energy transferred, via particle

collisions, from a region of high temperature to a

region of lower temperature.

• 1 calorie will raise the temperature of 1 gram of

water by 1 oC.

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• 4186 Joules are required to raise the temperature

of 1 kg of water by 1 oC.

1 calorie = 4.186 Joules

1 diet Calorie = 1000 calories

Specific Heat (Solids and Liquids) The heat energy Q that must be supplied or removed to change the temperature of a substance of mass m by an amount ΔT is

!

Q = mc"T (solids and liquids, one phase only)

where c = the specific heat (capacity) of the substance. Water has a relatively high value of specific heat, cwater = 4186 J/kg oC. One may also use:

!

c =1mdQdT

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Molar Specific Heats (gases) For the case of gases, one introduces the molar specific heat C as

!

Q = nC"T

where n is the number of moles of the substance. More

comments on gases in a later chapter. Also one may

define the molar specific heat as

!

C =1ndQdT

17.6 Calorimetry and Phase Changes Latent Heat L The Latent Heat L is the heat energy per kilogram that must be added or removed when a substance changes from one phase to another (at a constant temperature).

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A. Melting and Freezing:

!

Q = ±mLf (phase change)

where m = mass of the substance undergoing the phase change. Lf = latent heat of fusion = the energy required to break all the intermolecular bonds in one kilogram so as to convert the solid phase to the liquid phase. (positive sign for melting, negative sign for freezing). For water: Lf = 3.33x105 J/kg B. Vaporization and Condensation:

!

Q = ±mLv (phase change) where m = mass of the substance undergoing the phase change. Lv = latent heat of vaporization = the energy required to break all the intermolecular bonds in one kilogram so as to convert the liquid phase to the gas phase. (positive sign for vaporization, negative sign for condensation). For water: Lv = 2.26x106 J/kg

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C. Latent Heat of Combustion: Chemical reactions such as combustion are analogous to phase changes in that they involve definite quantities of heat. Complete combustion of 1 gram of gasoline produces about 46,000 J or about 11,000 calories. Thus the heat of combustion of gasoline is

Lc = 4.6x107 J/kg (combustion of gasoline)

which is about 46 million Joules per kilogram of gasoline. Energy values of foods are defined similarly. When we say that a gram of peanut butter “contains 6 calories”, we mean that 6 cal of heat (6000 cal or 25,000 J) is released when the carbon and hydrogen atoms in the peanut butter react with oxygen and are completely converted to CO2 and H2O.

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Equations of the type “heat gained = heat lost” Here both sides of the equation must have the same algebraic sign. Therefore, when calculating heat energy contributions, always write any temperature changes as the higher temperature minus the lower temperature! 17.7 Mechanisms of Heat Transfer: Conduction, Convention, and Radiation A. Conduction This is the process whereby heat energy is transferred

directly through a material, with any bulk motion of the

material playing no role in the energy transfer. Thermal

conduction takes place via:

(i) exchange of kinetic energy between atoms (collisions) (ii) motion of free electrons. The rate of energy transfer (“power” or “heat current”) H by thermal conduction is given by:

!

Power = H =k A Thot "Tcold( )

L=Q#t

in Watts

where

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k = thermal conductivity of the material A = cross-sectional area of the material L = length of the material (from hot end to cold end) Thot – Tcold = temp. difference between hot and cold ends • The thermal energy always travels from hot to cold.

• Thot and Tcold are maintained by reservoirs.

B. Convection This is the process in which the total energy is carried from place to place by the bulk movement of a fluid. There are convection currents in a pan of water being heated by a flame.

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C. Radiation This is the process whereby energy is transferred by means of electromagnetic waves. Every object emits electromagnetic radiation as a result of the thermal motion of its atoms or molecules. Stefan – Boltzmann Law of Radiation: The rate at which at object emits energy by thermal radiation is:

!

Poweremitted = H =Q"t

=# e AT 4

where != Stefan – Boltzmann constant = 5.67x10-8 W/m2K4 A = surface area of the object T = absolute temperature on surface of the object

!

e = emissivity, 0 ≤

!

e ≤ 1. This is equal to the fraction of the incident radiation that is absorbed by the surface.

Therefore, an object that is a good absorber is also a good emitter. Do not confuse emission with reflection! Absorption and Reflection are opposite processes. A good absorber reflects very little radiant energy. Hence a

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surface that reflects very little or no radiant energy looks dark. Similarly, poor absorbers are also good reflectors. Clean snow is a good reflector and therefore does not melt rapidly in sunlight. In the summer, light colored buildings stay cooler because they reflect much of the incoming radiation (good reflector thus poor absorber of radiation). In the winter, light colored building stay warmer because they are poor emitters (since poor absorber) so they retain much of their internal energy than darker buildings. ! paint your house white if you wish to conserve energy (and save $$$) ! If an object is at a temperature T and its surroundings are at a temperature Ts , then the net rate of energy transfer for the object as a result of thermal radiation is

!

Hnet = Hemitted "Habsorbed

!

Hnet =" eAT 4 #" eATs4

!

Hnet =" e A T 4 #To4( )

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When an object is in thermal equilibrium with its environment, it radiates and absorbs energy at the same rate, and so Hnet = 0 and its temperature remains constant! Thermos Bottles A double walled glass container with a vacuum between the walls. The glass walls are silvered to reflect radiation (sometimes these are called heat waves), and the vacuum prevents heat energy loss by conduction through the walls. If you keep the thermos bottle capped, then you prevent heat energy loss by convection.