Chapter 12



TEMPERATURE, HEAT AND GASES PART 1

PREVIEW

The total internal energy of the molecules of a substance is called thermal energy. The temperature of a substance is a measure of the average kinetic energy of the molecules in the substance, and gives an indication of how hot or cold the substance is relative to some standard. The energy transferred between two substances because of a temperature difference is called heat. Many substances expand when heated.

Please note that volume expansion is not included on the AP Physics B exam.

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Important Terms

absolute zero

the lowest possible temperature, at which all molecular motion would cease and a gas would have no volume.

calorie

the amount of heat required to raise the temperature of one gram of water by one Celsius degree

Celsius (C)

temperature scale in which the freezing point of water is 0 and the boiling point of water is 100 .

heat

the energy which is transferred from one body to another because of a temperature difference

Kelvin (absolute) temperature scale

scale in which zero Kelvins is defined as absolute zero, the temperature at which all molecular motion ceases

temperature

the property of a body which indicates how hot or cold a substance is with respect to a standard

thermal energy

the sum of the internal potential and kinetic energy of the random motion of the molecules making up an object

thermal equilibrium

state between two or more objects in which temperature doesn’t change

thermal expansion

increase in length or volume of a material due to an increase in temperature

Equations and Symbols

[pic]

where

TK = Kelvin temperature

TC = Celsius temperature

ΔT = change in temperature

ΔL = change in length

Lo = initial length

α = coefficient of linear expansion

Common Temperature Scales and The Kelvin Temperature Scale

Temperature is the measure of how hot or cold a substance is relative to some standard. It is the measure of the average kinetic energy of the molecules in a substance. The two temperature scales used most widely in scientific applications is the Celsius scale and the Kelvin scale. The only difference between them is where each starts. On the Celsius scale, the freezing point of water is 0( C, and the boiling point of water (at standard pressure) is 100( C. The Kelvin scale has temperature units which are equal in size to the Celsius degrees, but the temperature of 0 Kelvin is absolute zero, defined as the temperature at which all molecular motion in a substance ceases. Zero Kelvin is equal to - 273.15( C, so we can convert between the Kelvin scale and the Celsius scale by the equation

TK = TC + 273

Note that we have rounded 273.15 to 273. The boiling point of water in Kelvin would be TK = 100( C + 273 = 373 K.

Linear Thermal Expansion

When a solid is heated, it typically expands. Different substances expand at different rates, which is why you might heat the lid of a jar when the lid is too tight. The metal lid will expand more than the glass jar when it is heated, making it easier to loosen. Solids undergo two types of expansion when heated: linear thermal expansion, which is the increase in any one dimension of the solid, and volume thermal expansion, which results in an increase in the volume of the solid. Volume expansion is not typically covered on

the AP Physics B exam. In the case of linear expansion, the change in length (L is proportional to the original length Lo and the change in temperature (T of the solid:

[pic]

where ( is the coefficient of linear expansion.

Example 1 The ends of a copper bar and a steel bar, each of length 0.20 m, are separated by a gap of 0.50 mm, as shown. The other ends of the bars are attached to a rigid frame which does not expand significantly when heated. If the two bars are heated from 0( C to 100( C, determine whether or not the bars will come into contact with each other.

Solution

Since the frame is rigidly attached to the floor, we can assume the expansion of each bar is toward the other bar. Finding the change in length of each bar:

Copper:

[pic]

Steel:

[pic]

Adding the two changes in length, we get 5.8 x 10-4 m = 0.58 mm. Thus the 0.50 mm gap will close and the bars will come into contact with each other.

Heat and Internal Energy

In any state of matter, the molecules are moving and therefore have energy. They have potential energy because of the bonds between them and kinetic energy because the molecules have mass and speed. The sum of the potential and kinetic energies of the molecules in a substance is called the internal energy of the substance. When a warmer substance is brought in contact with a cooler substance, some of the kinetic energy of the molecules in the warmer substance is transferred to the cooler substance. The energy

representing the kinetic energy of molecules that is transferred spontaneously from a warmer substance to a cooler substance is called heat energy. Heat is generally given the symbol Q, and, since it is a form of energy, is measured in Joules (J) or calories (cal).

REVIEW QUESTIONS

For each of the multiple choice questions below, choose the best answer.

1. The average kinetic energy of the molecules in a substance is most closely associated with

(A) heat

(B) temperature

(C) expansion

(D) absolute zero

(E) potential energy

2. The Celsius temperature at absolute zero is equal to

(A) 0( C

(B) 100( C

(C) 273( C

(D) – 273( C

(E) – 100( C

3. Which of the following is true of the Celsius and Kelvin temperature scales?

(A) Both the Celsius and Kelvin

temperature scales have negative

values.

(B) A Kelvin degree and a Celsius

degree are equivalent in size.

(C) A Kelvin degree is larger in size

than a Celsius degree.

(D) A Kelvin degree is smaller in size

than a Celsius degree.

(E) The Kelvin scale reaches much

higher temperatures than the Celsius

scale.

4. In general, when a solid is heated, it

(A) expands proportionally to the change

in temperature

(B) contracts proportionally to the

change in temperature

(C) expands inversely proportionally to

the change in temperature

(D) contracts inversely proportionally to

the change in temperature

(E) does not expand nor contract.

5. A brass spring has a spring constant k. When the spring is heated, the spring constant will

(A) increase

(B) decrease

(C) remain the same

(D) increase, then decrease

(E) decrease then increase

6. Which of the following statement(s)

is/are true?

I. Every substance contains heat.

II. For heat to flow between two

substances, they must be at different

temperatures.

III. The internal energy of a substance is

equal to the kinetic energy of the

molecules in the substance.

(A) I and II only

(B) II and III only

(C) II only

(D) III only

(E) I, II, and III

ANSWERS AND EXPLANATIONS TO REVIEW QUESTIONS

Multiple Choice

1. B

The temperature of a substance is proportional to the average kinetic energy of its molecules.

2. D

TC = TK – 273 = 0 – 273 = – 273( C

3. B

The Kelvin and Celsius degrees are equivalent in size, they are simply offset by 273.

4. A

In the equation [pic], the coefficient of linear expansion α and the initial length of the metal Lo are both constants. Thus the change in length is proportional to the change in temperature.

5. B

A heated spring will lengthen, causing the spring to be less stiff, and the spring constant to decrease.

6. C

Heat can only be transferred between substances of different temperatures. It is not proper to say that a substance contains heat, but heat is the energy transferred between two substances. The internal energy is the sum of the kinetic and potential energy of the molecules in a substance.

PART 2: THE TRANSFER OF HEAT

PREVIEW

Heat can be transferred by conduction, convection, or radiation. Conduction is the transfer of heat through a material like a solid, in which there is no bulk movement of the material. Convection is the transfer of heat through materials such as liquids or gases. Radiation is heat transfer by electromagnetic waves.

QUICK REFERENCE

Important Terms

blackbody

a material which is a perfect absorber of heat, and also a perfect emitter of heat

convection

heat transfer by the movement a heated substance, such as currents in a fluid

conduction

heat transfer through a material, such as a solid, without bulk movement of the material

radiation

the transmission of energy by electromagnetic waves

thermal conductor

a material through which heat can easily flow

thermal insulator

a material that conducts heat poorly

Equations and Symbols

No equations from this work on heat are included on the AP Physics B exam.

DISCUSSION OF SELECTED SECTIONS

There are three ways of transferring heat from one place to another:

Convection

Convection is the transfer of heat by the bulk movement of a fluid. If the air near the floor of a cool room is heated, it expands and becomes less dense than the air above it, causing it to rise. As it rises, it cools, becomes more dense again, and falls toward the floor. If the air near the floor is continually heated, the cycle will repeat itself. Water heated in a pan is an example of heat transfer by convection, since water near the bottom of the pan near the fire is heated, rises, cools, then falls again. If the temperature gets high enough, the water begins to boil as it cools itself by transferring heat to the air by convection.

Conduction

Conduction is the transfer of heat directly through a material, or by actual contact between two materials. Metals are typically good heat conductors. In fact, materials which are good electrical conductors are usually good heat conductors as well. A material which is not a good heat conductor, like wood or air, is called an insulator. If you place an iron skillet on a fire, heat is transferred by conduction to the handle of the skillet. If you grasp the iron handle with your bare hand, you will feel it transfer heat to your hand by conduction.

Radiation

Radiation is the process by which heat is transferred by electromagnetic waves. We receive heat from the sun by radiation principally in the form of light, infrared, and ultraviolet waves. Microwave ovens use microwaves to transfer heat to food. And if you stand near a roaring campfire, you will feel the heat radiating from the fire in the form of light and infrared rays.

REVIEW QUESTIONS

For each of the multiple choice questions below, choose the best answer.

1. Cooking oil is poured into an iron pan which is heated over a flame. The heated oil begins rising to the top. The order of heat transfers during the entire process is

(A) conduction, convection, radiation

(B) convection, conduction, radiation

(C) radiation,convection,conduction

(D) conduction, radiation, convection

(E) radiation, conduction, convection,

2. Gases in the sun are heated and rise to the surface. A boy picks up a wrench which has been lying in the hot sun on a summer day. The order of heat transfers during the entire process is

(A) conduction, convection, radiation

(B) convection, radiation, conduction

(C) radiation,convection,conduction

(D) conduction, radiation, convection

(E) radiation, conduction, convection,

3. The air in a hair dryer is heated by

(A) convection

(B) conduction

(C) radiation

(D) insulation

(E) temperature

4. As water boils, the heat transfer through the water is best described as

(A) convection

(B) conduction

(C) radiation

(D) insulation

(E) temperature

5. Old houses were often built with high ceilings in the rooms so that they would be cooler in the warmer months. This was to take advantage of

(A) convection

(B) conduction

(C) radiation

(D) insulation

(E) temperature

ANSWERS AND EXPLANATIONS TO REVIEW QUESTIONS

Multiple Choice

1. E

The flame heats the pan by radiation, heat is transferred through the pan by conduction, and heat rises through the oil by convection.

2. B

Hot gases rise to the surface of the sun by convection, heat is transferred through empty space by radiation, and heat is transferred from the wrench to the boy’s hand by conduction.

3. C

The air is heated by a hot, glowing coil of wire which emits radiation.

4. A

The water is heated and rises to the top where it cools and sinks to the bottom again.

5. A

Warm air near the floor rises to the ceiling, leaving the space near the floor cooler.

PART 3: THE IDEAL GAS LAW AND KINETIC THEORY

PREVIEW

Kinetic molecular theory involves the study of matter, particularly gases, as very small particles in constant motion. Because of the motion of the particles, an ideal gas has internal energy that can be transferred. We study gases by relating their pressure, volume, number of moles, and temperature in the ideal gas law.

QUICK REFERENCE

Important Terms

atomic mass unit

one-twelfth the mass of a carbon-12 atom

ideal gas law

the law which relates the pressure, volume, number of moles, and temperature of an ideal gas

internal energy

the sum of the potential and kinetic energy of the molecules of a substance

kinetic theory of gases

the description of matter as being made up of extremely small particles which are in constant motion

mole

one mole of a substance contains Avogadro’s number (6.02 x 1023) of molecules or atoms

Equations and Symbols

[pic]

where:

P = pressure

V = volume

n = number of moles

R = universal gas constant

= 8.31 J / (mol K)

KEavg = average kinetic energy of

molecules

vrms = root-mean-square speed

kB = Boltzmann constant

= 1.38 x 10-23 J/K

T = Kelvin temperature

M = molecular mass

μ = mass of molecule

DISCUSSION OF RELEVANT MATERIAL

The Mole, Avogadro’s Number, and Molecular Mass

When we are dealing with small particles like atoms and molecules, it is convenient to express their masses in atomic mass units (u) rather than kilograms. Atomic mass units and kilograms are related by the conversion 1 u = 1.6605 x 10-27 kg, which is approximately the size of a proton. When we buy 12 eggs we say we have a dozen eggs, but if we have 6.022 x 1023 atoms, we say we have a mole of those atoms. In other words,

a mole of a substance is Avogadro’s number (6.022 x 1023) of atoms or molecules of that substance.

The Ideal Gas Law

All gases display similar behavior. When examining the behavior of gases under varying conditions of temperature and pressure, it is most convenient to treat them as ideal gases. An ideal gas represents a hypothetical gas whose molecules have no intermolecular forces, that is, they do not interact with each other, and occupy no volume. Although gases in reality deviate from this idealized behavior, at relatively low pressures and high temperatures many gases behave in nearly ideal fashion. Therefore, the assumptions used for ideal gases can be applied to real gases with reasonable accuracy.

The state of a gaseous sample is generally defined by four variables:

• pressure (p),

• volume (V),

• temperature (T), and

• number of moles (n),

though as we shall see, these are not all independent. The pressure of a gas is the force per unit area that the atoms or molecules exert on the walls of the container through collisions. The SI unit for pressure is the pascal (Pa), which is equal to one newton per meter squared. Sometimes gas pressures are expressed in atmospheres (atm). One atmosphere is equal to 105 Pa, and is approximately equal to the pressure the earth’s atmosphere exerts on us each day. Volume can be expressed in liters (L) or cubic meters (m3), and temperature is measured in Kelvins (K) for the purpose of the gas laws. Recall that we can find the temperature in K by adding 273 to the temperature in Celsius. Gases are often discussed in terms of standard temperature and pressure (STP), which refers to the conditions of a temperature of 273 K (0(C) and a pressure of 1 atm.

These four variables are related to each other in the ideal gas law:

[pic]

where R is a constant known as the universal gas constant = 8.31 J / (mol K). If the number of moles of a gas does not change during a process, then n and R are constants, and we can write the equation as the combined gas law:

[pic]

where a subscript of 1 indicates the state of the gas before something is changed, and the subscript 2 indicates the state of the gas after something is changed.

Example 1

A cylinder is closed at one end with a piston which can slide to change the closed volume of the cylinder. When the piston is at the end of the cylinder, as in Figure III, the volume of the cylinder is 1.0 liter. The area of the piston is 0.01 m2. The piston is positioned at half the length of the cylinder in Fig. I, and the cylinder is filled with an ideal gas. The force F necessary to hold the piston in this position is 10 N, and the temperature of the gas is 50˚ C.

(a) Determine the following for the gas in Fig. I:

i. the volume of the gas

ii. the pressure of the gas

A force is applied to the piston so that it is now positioned at one-third the length of the cylinder from its closed end, but the temperature of the gas remains at 50˚ C.

(b) Determine the following for the gas in Fig. II:

i. the volume of the gas

ii. the pressure of the gas

(c) The temperature of the gas is raised to 80˚ C between Fig. II and Fig. III. Determine the pressure of the gas in Fig. III.

Solution

(a) i. The volume of the gas is half of the full cylinder, or 0.5 liter.

ii. [pic]

(b) i. The volume of the gas is one-third of the full cylinder, or 0.33 liter.

ii. At a constant temperature, the pressure and volume of the gas are inversely proportional according to Boyle’s law:

[pic]

(c) Converting Celsius degrees to Kelvins:

[pic]

Kinetic Theory of Gases

As indicated by the gas laws, all gases show similar physical characteristics and behavior. A theoretical model to explain why gases behave the say they do was developed during the second half of the 19th century. The combined efforts of Boltzmann, Maxwell, and others led to the kinetic theory of gases, which gives us an understanding of gaseous behavior on a microscopic, molecular level. Like the gas laws, this theory was developed in reference to ideal gases, although it can be applied with reasonable accuracy to real gases as well.

The assumptions of the kinetic theory of gases are as follows:

• Gases are made up of particles whose volumes are negligible compared to the container volume.

• Gas atoms or molecules exhibit no intermolecular attractions or repulsions.

• Gas particles are in continuous, random motion, undergoing collisions with other particles and the container walls.

• Collisions between any two gas particles are elastic, meaning that no energy is dissipated and kinetic energy is conserved.

• The average kinetic energy of gas particles is proportional to the absolute (Kelvin) temperature of the gas, and is the same for all gases at a given temperature. As listed in the list of equations, the average kinetic energy of each molecule is related to Kelvin temperature T by the equation

[pic], where kB is the Boltzmann constant, 1.38 x 10-23 J/K. The root-mean

square speed of each molecule can be found by [pic], where ( is the mass of each molecule. This equation is very seldom used on the AP Physics B exam, and is provided on the exam if needed.

Example 2

The temperature of an ideal gas is 60˚ C.

(a) Find the average kinetic energy of the molecules of the gas.

(b) On the axes below, sketch a graph of

i. average kinetic energy Kavg vs Kelvin temperature T

ii. root-mean-square speed vrms of each molecule in the gas vs. Kelvin temperature T.

Solution

(a) [pic]

(b) The average kinetic energy of each molecule is directly proportional to the Kelvin temperature of the gas, and vrms is proportional to the square root of the Kelvin temperature of the gas:

REVIEW QUESTIONS

For each of the multiple choice questions below, choose the best answer.

1. A mole is to Avogadro’s number as

(A) kinetic energy is to temperature

(B) atomic mass unit is to kg

(C) gas is to liquid

(D) pressure is to volume

(E) decade is to ten years

2. Which of the following is NOT true of an ideal gas?

A) Gas molecules have no intermolecular forces.

B) Gas particles are in random motion.

C) Gas particles have no volume.

D) The collisions between any two gas particles are elastic.

E) The average kinetic energy of the gas molecules is proportional to the temperature in Celsius degrees.

3. A sample of argon occupies 50 liters at standard temperature. Assuming constant pressure, what volume will argon occupy if the temperature is doubled?

A) 25 liters

B) 50 liters

C) 100 liters

D) 200 liters

E) 2500 liters

4. What is the final pressure of a gas that expands from 1 liter at 10(C to 10 liters at 100(C if the original pressure was 3 atmospheres?

A) 0.3 atm

B) 0.4 atm

C) 3 atm

D) 4 atm

E) 30 atm

5. Which of the following pressure vs. volume graphs best represents how pressure and volume change when temperature remains constant?

(A)

(B)

(C)

(D)

(E)

6. Which of the following volume vs. temperature graphs best represents how volume changes with Kelvin temperature if the pressure remains constant?

(A)

(B)

(C)

(D)

(E)

Free Response Question

Directions: Show all work in working the following question. The question is worth 10 points, and the suggested time for answering the question is about 10 minutes. The parts within a question may not have equal weight.

1. (10 points)

A special balloon contains 3 moles of an ideal gas and has an initial pressure of 3 x 105 Pa and an initial volume of 0.015 m3.

(a) Determine the initial temperature of the gas.

(b) The pressure in the balloon is changed in such a way as to increase the volume of the balloon to 0.045 m3 but the temperature is held constant. Determine the pressure of the gas in the balloon at this new volume.

(c) If the balloon contracts to one-third of its initial volume, and the pressure is increased to twice its initial value, describe the change in temperature that would have to take place to achieve this result.

ANSWERS AND EXPLANATIONS TO REVIEW QUESTIONS

Multiple Choice

1. E

A mole is defined as a certain number of things (6 x 1023), and a decade is a certain number of years (10).

2. E

All of the statements are true, except for “Celsius” should be replaced with “Kelvin”.

3. C

With constant pressure, volume and temperature are proportional to each other, so twice the temperature would result in twice the volume.

4. B

First, we must convert Celsius to Kelvin:

T1 = 10˚ C + 273 = 283 K

T2 = 100˚ C + 273 = 373 K

P1 = 3 atm

V1 = 1 l

V2 = 10 l

[pic]

Solving for P2 and substituting, we get P2 = 0.4 atm.

5. E

Pressure and volume are inversely proportional to each other at constant temperature.

6. A

Volume and temperature are proportional to each other at constant temperature.

Free Response Question Solution

(a) 4 points

[pic]

(b) 3 points

For a constant temperature,

[pic]

(c) 3 points

[pic]

In order to keep the ratios constant on both sides,

[pic]

Thus, the temperature would have to decrease to 1/6 its initial value for these changes in the pressure and volume to take place.

-----------------------

Figure not drawn to scale

Gap

F

A

Fig. I Fig. II Fig. III

Kavg

T

vrms

T

Kavg

T

vrms

T

P

V

P

V

P

V

P

V

P

V

V

T

V

T

V

T

V

T

V

T

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