Lecture 23, Nov. 19

Physics 207: Lecture 23, Pg 1

Lecture 23, Nov. 19Goals:Goals:

• Chapter 17Chapter 17 Apply heat and energy transfer processes Recognize adiabatic processes

• Chapter 18Chapter 18 Follow the connection between temperature, thermal energy, and the average translational kinetic energy molecules Understand the molecular basis for pressure and the ideal-gas law. To predict the molar specific heats of gases and solids.

• AssignmentAssignment HW10, Due Sunday (11:59 PM) For Wednesday, Read through all of Chapter 18


Exam III Room assignments

613 Room 2223 Koki

601 Room 2241 Matt603 Room 2241 Heming608 Room 2241 Matt609 Room 2241 Heming607 Room 2241 Koki

And all others in Room 2103602 604605606610611612614


Work and Ideal Gas Processes (on system)

Isothermal

)/VV( nRT ifnW

Isobaric

)V-V( p ifW Isochoric

0W

)( 12constconst

2

1

2

1

VVPdVW V

VVV V

dV

V

FYI: Adiabatic (and reversible)


Heat and Ideal Gas Processes (on system)

Isothermal Expansion/Contraction

WQ Isobaric

Isochoric

TnCQ V

TRCnTnCQ Vp )(

Adiabatic

0Q


Identify the nature of paths 1, 2, 3 and 4

(A) Isobaric

(B) Isothermal

(C) Isochoric

(D) Adiabatic

Exercise Identify processes

p

V

1

2

3

4

T1

T2

T3T4


Two process are shown that take an ideal gas from state 1 to

state 3.

Compare the work done by process A to the work done by

process B.

A. WA > WB

B. WA < WB C. WA = WB = 0D. WA = WB but neither is zero

ON BYA 1 3 W12 = 0 (isochoric)B 1 2 W12 = -½ (p1+p2)(V2-V1) < 0 -W12 > 0B 2 3 W23 = -½ (p2+p3)(V1-V2) > 0 -W23 < 0B 1 3 = ½ (p3 - p1)(V2-V1) > 0 < 0


Heat and Latent Heat

Latent heat of transformation L is the energy required for 1 kg of substance to undergo a phase change. (J / kg)

Q = ±ML Specific heat c of a substance is the energy required to raise the

temperature of 1 kg by 1 K. (Units: J / K kg )

Q = M c ΔT

Molar specific heat C of a gas at constant volume is the energy required to raise the temperature of 1 mol by 1 K.

Q = n CV ΔT


Most people were at least once burned by hot water or steam. Assume that water and steam, initially at 100°C, are cooled down

to skin temperature, 37°C, when they come in contact with your skin. Assume that the steam condenses extremely fast, and that the specific heat c = 4190 J/ kg K is constant for both liquid water and steam.

Under these conditions, which of the following statements is true?

(a) Steam burns the skin worse than hot water because the thermal conductivity of steam is much higher than that of liquid water.

(b) Steam burns the skin worse than hot water because the latent heat of vaporization is released as well.

(c) Hot water burns the skin worse than steam because the thermal conductivity of hot water is much higher than that of steam.

(d) Hot water and steam both burn skin about equally badly.

Exercise Latent Heat


Exercise Latent Heat Most people were at least once burned by hot water or steam. Assume

that water and steam, initially at 100°C, are cooled down to skin temperature, 37°C, when they come in contact with your skin. Assume that the steam condenses extremely fast, and that the specific heat c = 4190 J/ kg K is constant for both liquid water and steam.

Under these conditions, which of the following statements is true?

(b) Steam burns the skin worse than hot water because the latent heat of vaporization is released as well.

How much heat H1 is transferred to the skin by 25.0 g of steam?

The latent heat of vaporization for steam is L = 2256 kJ/kg.

H1 = 0.025 kg x 2256 kJ/kg = 63.1 kJ How much heat H2 is transferred to the skin by 25.0 g of water?

H2 = 0.025 kg x 63 K x 4190 J/ kg K = 6.7 kJ


Energy transfer mechanisms Thermal conduction (or conduction) Convection Thermal Radiation

For a material of cross-section area A and length L, spanning a temperature difference ΔT = TH – TC, the rate

of heat transfer is

where k is the thermal conductivity, which characterizes whether the material is a good conductor of heat or a poor conductor.

Q / t = k A T / x


Energy transfer mechanisms

Thermal conduction (or conduction): Energy transferred by direct contact. e.g.: energy enters the water through

the bottom of the pan by thermal conduction.

Important: home insulation, etc.

Rate of energy transfer ( J / s or W ) Through a slab of area A and

thickness x, with opposite faces at different temperatures, Tc and Th

Q / t = k A (Th - Tc ) / x

k :Thermal conductivity (J / s m °C)


Thermal Conductivities

Aluminum 238 Air 0.0234 Asbestos 0.25

Copper 397 Helium 0.138 Concrete 1.3

Gold 314 Hydrogen 0.172 Glass 0.84

Iron 79.5 Nitrogen 0.0234 Ice 1.6

Lead 34.7 Oxygen 0.0238 Water 0.60

Silver 427 Rubber 0.2 Wood 0.10

J/s m °C J/s m °C J/s m °C


100 C

Two thermal conductors (possibly inhomogeneous) are butted together and in contact with two thermal reservoirs held at the temperatures shown.

Which of the temperature vs. position plots below is most physical?

Exercise Thermal Conduction

300 C

Position

Te

mp

era

ture

Position

Te

mp

era

ture

Position

Te

mp

era

ture

(A) (B) (C)


Energy transfer mechanisms Convection:

Energy is transferred by flow of substance

1. Heating a room (air convection)

2. Warming of North Altantic by warm waters from the equatorial regions

Natural convection: from differences in density Forced convection: from pump of fan

Radiation: Energy is transferred by photons

e.g.: infrared lamps Stefan’s Law

= 5.710-8 W/m2 K4 , T is in Kelvin, and A is the surface area e is a constant called the emissivity

P = A e T4 (power radiated)


Minimizing Energy Transfer

The Thermos bottle, also called a Dewar flask is designed to minimize energy transfer by conduction, convection, and radiation. The standard flask is a double-walled Pyrex glass with silvered walls and the space between the walls is evacuated.

VacuumVacuum

SilveredSilveredsurfacessurfaces

Hot orHot orcoldcoldliquidliquid


Anti-global warming or the nuclear winter scenario

Assume P/A = I = 1340 W/m2 from the sun is incident on a thick dust cloud above the Earth and this energy is absorbed, equilibrated and then reradiated towards space where the Earth’s surface is in thermal equilibrium with cloud. Let e (the emissivity) be unity for all wavelengths of light.

What is the Earth’s temperature?P = A T4= (4 r2) T4 = I r2 T = [I / (4 x )]¼ = 5.710-8 W/m2 K4 T = 277 K (A little on the chilly side.)


Ch. 18, Macro-micro connectionMolecular Speeds and Collisions

• A real gas consists of a vast number of molecules, each moving randomly and undergoing millions of collisions every second.

• Despite the apparent chaos, averages, such as the average number of molecules in the speed range 600 to 700 m/s, have precise, predictable values. • The “micro/macro” connection is built on the idea that the macroscopic properties of a system, such as temperature or pressure, are related to the average behavior of the atoms and molecules.


Molecular Speeds and Collisions

A view of a Fermi chopper


Molecular Speeds and Collisions


Mean Free Path

If a molecule has Ncoll collisions as it travels distance L, the average distance between collisions, which is called the mean free path λ (lowercase Greek lambda), is


Macro-micro connection Assumptions for ideal gas:

# of molecules N is large They obey Newton’s laws Short-range interactions with

elastic collisions Elastic collisions with walls

(an impulse…..pressure)

What we call temperature T is a direct measure of the average translational kinetic energy

What we call pressure p is a direct measure of the number density of molecules, and how fast they are moving (vrms)

avg32 VN

p

avg32 Bk

T

m

Tkvv B

rms

3)( avg

2


Lecture 23, Nov. 19

• AssignmentAssignment HW10, Due Sunday (11:59 PM) For Wednesday, Read through all of Chapter 18

Following slides are for Wednesday


Kinetic energy of a gas

The average kinetic energy of the molecules of an ideal gas at 10°C has the value K1. At what temperature T1 (in degrees Celsius) will the average kinetic energy of the same gas be twice this value, 2K1?

(A) T1 = 20°C

(B) T1 = 293°C

(C) T1 = 100°C

The molecules in an ideal gas at 10°C have a root-mean-square (rms) speed vrms.

At what temperature T2 (in degrees Celsius) will the molecules have twice the rms speed, 2vrms?

(A) T2 = 859°C

(B) T2 = 20°C

(C) T2 = 786°C

avg2

23

21 Tkvm B


Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2.

1. 1. If T is doubled, what happens to the rate at which If T is doubled, what happens to the rate at which a single a single moleculemolecule in the gas has a wall bounce? in the gas has a wall bounce?

(B) x2(A) x1.4 (C) x4

22. If N is doubled, what happens to the rate at which . If N is doubled, what happens to the rate at which a a single moleculesingle molecule in the gas has a wall bounce? in the gas has a wall bounce?

(B) x1.4(A) x1 (C) x2

Exercise

PV NkBT

1

2mv 2

3

2kBT


Degrees of freedom or “modes” Degrees of freedom or “modes of energy storage in the system” can

be: Translational for a monoatomic gas (translation along x, y, z axes, energy stored is only kinetic) NO potential energy

Rotational for a diatomic gas (rotation about x, y, z axes, energy stored is only kinetic)

Vibrational for a diatomic gas (two atoms joined by a spring-like molecular bond vibrate back and forth, both potential and kinetic energy are stored in this vibration)

In a solid, each atom has microscopic translational kinetic energy and microscopic potential energy along all three axes.


Degrees of freedom or “modes”

A monoatomic gas only has 3 degrees of freedom (just K, kinetic)

A typical diatomic gas has 5 accessible degrees of freedom at room temperature, 3 translational (K) and 2 rotational (K)

At high temperatures there are two more, vibrational with K and U

A monomolecular solid has 6 degrees of freedom

3 translational (K), 3 vibrational (U)


The Equipartition Theorem

The equipartition theorem tells us how collisions distribute the energy in the system. Energy is stored equally in each degree of freedom of the system.

The thermal energy of each degree of freedom is:

Eth = ½ NkBT = ½ nRT A monoatomic gas has 3 degrees of freedom

A diatomic gas has 5 degrees of freedom

A solid has 6 degrees of freedom

Molar specific heats can be predicted from the thermal energy, because

nRTEth 23

nRTEth 25

TnCEth nRTCV 2

3

gas Monoatomic

nRTCV 25

gas Diatomic

nRTCV 3

solid Elemental

nRTEth 3


Exercise A gas at temperature T is mixture of hydrogen and helium gas.

Which atoms have more KE (on average)?

(A) H (B) He (C) Both have same KE

How many degrees of freedom in a 1D simple harmonic oscillator?

(A) 1 (B) 2 (C) 3 (D) 4 (E) Some other number

Lecture 23, Nov. 19

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Transcript of Lecture 23, Nov. 19