Chapter 17 – Temperature and Heat

I. Introduction – We will examine two types of interactions that take place between a system and the environment (everything except the system). The two interactions are the transfer of heat (disordered energy) between the system and the environment, and the work done by or on the system by the environment.

II. Two approaches when examining the interactions

A. Microscopic – statistical mechanics

B. Macroscopic – thermodynamics

C. We will then examine the relationship between the two approaches

III. Start with temperature, T

A. Definition: a measure of the hotness or coldness of a body. T turns out to be proportional to the average translational kinetic energy of the molecules.

B. Observation – hot and cold objects placed in an adiabatic container, after a time the temperature sensation will be the same – reach “thermal equilibrium.”

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TH TC Teq Teq

TH > TC TH Teq TC

environment

system

C. Consequence of observation:

Perform the experiment with three objects: A, B, C. If TA = TC and TB = TC , then TA = TB . This referred to as the “Zeroth law of thermodynamics”

The object C is equivalent to something that measures the temperature called a “thermometer”

D. Thermometers – all possess a property that varies with temperature called the “thermometric property”

1. touch

2. mercury (alcohol) in glass thermometer

3. thermocouple

4. optical pyrometer

5. constant volume gas thermometer

E. Temperature Scales and Calibration

1. Common temperature scales: Fahrenheit (oF) and Celsius (oC)

2. Calibration: need fixed points at which values of temperature are defined. Take ice point and steam point of water as two fixed points that are used to calibrate thermometers.

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A C B C

steam point

ice point

Fahrenheit Celsius

3. A constant volume gas thermometer containing a gas under low pressure effectively produces a linear relationship between pressure and temperature.

This allows us to define other temperature scales: Kelvin or absolute (K) and Rankine (oR). In both of these cases, the lowest temperature is called zero degrees Rankine or 0 Kelvin.

Kelvin (K):

Rankine (oR):

4. Look at the relationship between the various temperature scales

Celsius Fahrenheit Kelvin Rankine

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ice point

steam point100

0

-273.1

212

32

-459.67 0 0

273.15

373.15

491.67

671.67

pressure

T (oC or oF)

V. Thermal Expansion

A. When heat is added to a material, the atoms or molecules vibrate more violently and the average distance between the molecules increases. This corresponds in most cases to an expansion of the material when heated.

B. One-dimensional expansion

C. Two-dimensional expansion

Does a hole cut in a plate get larger or smaller when the plate is heated?

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L

L

Temp T

Temp T + T

L

W

L

WTemp T

Temp T + T

W

L

U

r

D. Three-dimensional expansion

E. Example: A steel container with a volume of 500 cm3 is filled with alcohol at 20oC. If the temperature of the container and alcohol are increased to 40oC, then does alcohol spill out or does the level of alcohol drop? Take steel = 1.1 x 10-5 (oC)-1 and alcohol = 1 x 10-3 (oC)-1 .

VI. Expansion and Contraction of Water

A. Variation of density with temperature

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T(oC)

(g/cm3) (g/cm3)

T(oC)0.90

1.00

0.95

0.9998

1.0000

0.9999

0 100 0 84

B. Why the anomalous behavior?

IV. Temperature and Heat

A. How can you change the temperature of an object? Add heat or take away heat. What is heat?

B. Examine what happens when two objects at different temperatures are confined to an adiabatic container. One is at temperature T1 and the other at temperature T2 , where T1

> T2 .

C. How does this happen?

D. Units of heat (energy units)

Joules (J)

calorie (cal)

kilocalorie (kcal)

British Thermal Unit (Btu)

Conversion factors: 1 cal = 4.186 J

1 Btu = 252 cal = 1054 J

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T1 T2 TT T1 > T > T2

E. How do you know how much heat given up or absorbed by an object that undergoes a temperature change dT?

1. Amount heat transferred is directly related to the change in temperature:

2. Amount of heat transferred is directly related to amount of material (mass):

3. Write the relationship as an equation:

4. Define: c specific heat capacity = or,

specific heat capacity = c = .

units:

5. Rearrange expression to find heat supplied or removed to cause a temperature change:

dQ = mc dT,

or, for a finite change in temperature

.

6. If the specific heat is constant over the temperature change, then the heat transferred can be written as

Q = mc T = mc(T2 – T1) .

F. Specific heat values of some materials at 20oC

Material c (cal/g oC, kcal/kg oC, or Btu/lb oF)

c (J/kg K)Water 1.00 4186Ice 0.49 2050Lead 0.0305 128Copper 0.0923 386Brass 0.092 385Iron 0.107 448Glass 0.2 837Wood 0.33 1380

G. Examples

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1. Find the amount of heat required to raise the temperature of 0.5 L of water from 20oC to 50oC. Express your answer in calories and in Joules.

2. Find the amount of heat required to raise the temperature of 2.0 kg of a material from 20oC to 50oC. The specific heat of the material is c = 1200 + 40T, in J/kg Co. The temperature T is in degrees Celsius.

H. Heat exchanges

Because we know that heat is energy, if no energy exchanges take place between the system and the environment, then energy must be conserved within the isolated system. That is,

1. A 100 g glass beaker contains 200 g of water at 20oC (in equilibrium). Three hundred (300) grams of water at 80oC is added to the beaker. What is the final equilibrium temperature? Assume no loses or gains in heat between the system and the environment.

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2. A 100 g copper block is dropped into 200 g of water at 20oC. When equilibrium is attained, the temperature of the combination is 30oC. What was the initial temperature of the copper? Assume this procedure takes place within an adiabatic container.

I. Heat capacities

1. specific heat

2. molar specific heat (or molar heat capacity), C

, n = number of moles of material and M is the molecular mass

units: or

Note that the heat transferred is:

,

or, if C is constant

For solids and liquids, the pressure and volume changes do not alter the values of the specific heat (c) and the molar heat capacity (C) by much. However, the pressure and volume changes greatly affect the heat capacities of gases. So in these cases a subscript is added to indicate what quantity is kept constant during the heat exchange. The molar heat capacity for processes taking place at a constant pressure is Cp, and for processes at a constant volume is Cv.

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V. Phase Changes

A. We found that when heat is added to a system, the temperature changes. However, there are some situations where heat can be exchanged and no temperature change occurs. In this case, a phase change is occuring. That is, when a material changes state (solid, liquid, gas), heat is added or removed, but no temperature change occurs.

B. Look at ice changing to water changing to steam

C. Heat of transformation is the amount of heat added or removed from a material in order to change that material from one state to another.

For water:

Heat of Fusion, Lf = 79.7 cal/g = 3.33 x 105 J/kg

Heat of Vaporization, Lv = 539 cal/g = 2.256 x 106 J/kg

D. Heat transferred during phase changes – look at it on the molecular level:

10

100o

C

0oC Heat added

-20oC

Temp, T

E. Examples

1. How much energy is required to take 100 g of ice from –20oC to steam at 100oC?

2. How much ice at –10oC is needed to cool 400 g of water and a 100 g glass from 20oC to 10oC?

3. Ten grams of steam at 100oC are mixed with 60 g of ice at –20oC. What is the final equilibrium temperature?

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VI. Heat Transfer – heat can be transferred from one place to another. Three methods:

Conduction – transfer of energy by molecular collisions

Convection – transfer of energy by the actual transport of the material

Radiation – transfer of energy by radiation

A. Conduction

Consider a slab of material of thickness dx, cross sectional area A, with a temperature difference of dT, and an amount of heat dQ passing through the slab in a time dt.

Deduce the heat current, H (the rate of heat flow), through the slab:

The heat current:

or,

,

where k is called the thermal conductivity.

k is large for a good conductor of heat and small for a poor conductor of heat:

For silver: k = 427 W/m K and for wood k = 0.08 W/m K

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dx

A

dT

dQ

B. Look at examples of “steady state” heat flow.

1. Given a rod of length L, cross sectional area A, thermal conductivity k, and temperatures T1 and T2 at the ends of the rod where T1 > T2. Let k be the value of the thermal conductivity of the rod.. Find the heat current, H. Assume that no heat is lost through the sides of the rod.

2. Two rods of lengths L1 and L2 , temperatures T1 and T2 at their ends, thermal conductivities k1 and k2 , and have the same cross sectional areas A,. Find H and the temperature at the junction.

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L

A

T1 T2

L1 L2

A k1 k2

T1 T T2

3. A hollow sphere has an inner radius R1 and outer radius R2. The corresponding temperatures are T1 and T2, respectively. Let T1 > T2. Find H.

B. Convection

Examples:

1. Water in a saucepan

2. Convection currents in the earth.

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R1

R2

Crust

Mantle

Outer core

Inner Core

3. Onshore and offshore breezes.

C. Radiation

Rate of flow of radiation energy leaving an object is given by the Stefan-Boltzmann Law:

H = eAT4 ,

Where H is the heat current or the rate of energy loss in Watts, A is the surface area in m2, T is the temperature of the object in Kelvins and is called the Stefan-Boltzmann constant = 5.67 x 10-8 W/m2 K4. e is called the emissivity of the material and depends on the transmission and reflection properties of the surface of the material.

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shore water

shore water