Chapter 13: Temperature and the Ideal Gas

PHYSICS 149: Lecture 25• Chapter 12: Sound

– 12.4 Standing Sound Waves

• Chapter 13: Temperature and the Ideal Gas– 13.1 Temperature– 13.2 Temperature Scales

Lecture 25 Purdue University, Physics 149 1

Final Exam• Wednesday, December 15, 8:00 – 10:00 AM• Place: MSEE B012• Chapters 1 – 15 (only the sections we covered)• The exam is closed book.• The exam is a multiple-choice test.• There will be 30 multiple-choice problems.

– Each problem is worth 10 points.

• You may make a single crib sheet.– you may write on both sides of an 8.5” × 11.0” sheet.

• Conflicts: I need to know by Monday Nov 29.• Adaptive learners should contact Prof. Neumeister ASAP

Lecture 10 2Purdue University, Physics 149


Two strings, one thick and the other thin, are connected to form one long string. A wave travels along the string and passes the point where the two strings are connected. Which of the following does not change after crossing that point:

A) frequencyB) wavelengthC) propagation speed

ILQ 1

ILQ 2Standing waves are produced by the superposition of two waves with

a) the same amplitude and direction of propagation, but different frequencies.

b) the same amplitude, frequency, and direction of propagation.

c) the same amplitude and frequency, and opposite propagation directions.

d) the same amplitude, different frequencies, and opposite directions of propagation.


Standing Waves• Standing waves occur when a wave is reflected at a boundary in

such a way that the wave appears to stand still.• In a standing wave on a string, every point moves (as a whole) in

simple harmonic motion (SHM) with the same frequency.• Every point reaches its maximum amplitude simultaneously, and

every point also reaches its minimum amplitude (namely, zero) simultaneously as well.

• Nodes are points of zero amplitude (that is, points that nevermove); antinodes are points of maximum amplitude. The distance between two adjacent nodes is ½ λ.

• Fixed end of a string is always a node, because it never moves.



• The natural frequency is related to the length of the string L. The lowest frequency (first harmonic) has one antinode

• The second harmonic has two antinodes

• The n-th harmonic

Standing Waves

Sound Waves• Sound is a longitudinal wave, with

compressions and rarefactions of air.

• The wave can be described by the gauge pressure p (a measure of the compression and rarefaction of the air) as a function of time.

• The wave can also be described by the displacement s (= Δx) of an element of the air, which will oscillate around an equilibrium position (in the direction of energy transport). The restoring force is caused by the air pressure.

When p is zero (node), s is maximum or minimum (antinode)!



Speed of Sound

Frequencies of Sound Waves

• Humans with excellent hearing can hear frequencies from 20 Hz to 20 kHz (audible range).

• The terms infrasound and ultrasound are used to describe sound waves with frequencies below 20 Hz and above 20 kHz, respectively.

• Some animals (such as dogs and dolphins) can here ultrasound, while some others (such as elephants) can here infrasound.



Intensity and Loudness• Intensity is the power per unit area

– I = P / A– Units: Watts/m2

• For Sound Waves– I = p0

2 / (2 ρ v) (po is the pressure amplitude)

– Proportional to p02 (note: Energy goes as A2)

• Loudness (Decibels)– Loudness perception is logarithmic– Threshold for hearing I0 = 10-12 W/m2

– β = (10 dB) log10 ( I / I0)– β2 – β1 = (10 dB) log10(I2/I1)


Log10 Review

• Log10(1) = 0• Log10(10) = 1• Log 10(100) = 2• Log10(1,000) = 3

• Log10(10,000,000,000) = 10

logx2 = 2 log x

log10(10x ) = x


Decibels• Sound intensity level β

– β = (10 dB) log10 ( I / I0)– Units: Bels

– A ratio of 107 indicates a sound intensity of 7 bels or 70 decibels (dB)

• An intensity of 0 dB corresponds to the hearing threshold– Intensity increase by factor 10 → intensity

level adds 10 dB– Adding 3 dB to the intensity level doubles the

intensity (log102=0.3)

Alexander Graham Bell(1847-1922)


Decibels ILQ

If 1 person can shout with loudness 50 dB. How loud will it be when 100 people shout?

A) 52 dB B) 70 dB C) 150 dB

β100 – β1 = (10 dB) log10(I100/I1)β100 = 50 + (10 dB) log10(100/1)β100 = 50 + 20


Standing Sound Waves

Pipe open at both ends


Standing Sound Waves

Pipe open at one end

Standing Sound Waves• An open end of a pipe is always a displacement antinode,

because the pressure at the end cannot be deviated much from atmospheric pressure (a pressure node).

• A closed end of a pipe is always a displacement node, because the air near it cannot move beyond that rigid surface.


ILQ• An open pipe is closed at one end. Which one

is not a possible standing wave pattern for this pipe?

a) (a)b) (b)c) (c)d) (d)



Organ Pipe Example

A 0.9 m organ pipe (open at both ends) is measured to have it’s n=2 harmonic at a frequency of 382 Hz. What is the speed of sound in the pipe?

Pressure Node at each end.

λ = 2 L / n n=1,2,3..

λ = L for n=2 harmonic f = v / λ

v = f λ = (382 s-1 ) (0.9 m)

= 343 m/s

Example(a) What should be the length of an organ pipe, closed at

one end, if the fundamental frequency is to be 261.5 Hz? (Assume the temperature of air is 20˚C.)– The speed of sound in air at 20˚C

v = (331 + 0.606 × 20) m/s = 343 m/s– For a pipe closed at one end, fn = nv/(4L)

L = v/(4f1) = 343 m/s / (4 × 261.5 Hz) = 32.8 cm

(b) What is the fundamental frequency of the organ pipe of part (a) if the temperature drops to 0.0˚C?– The speed of sound in air at 20˚C

v = (331 + 0.606 × 0) m/s = 331 m/s– From part (a), L = 32.8 cm– For a pipe closed at one end, fn = nv/(4L)

f1 = 331 m/s / (4 × 0.328 m) = 252.4 Hz



Can You Tell Temperature

• NO

• Need Thermometers

Temperature• If heat can flow between two objects or systems, the objects or

systems are said to be in thermal contact.

• If two objects are placed in thermal contact, energy will be transferred from the one at higher temperature to the one at lower temperature.

• The energy that flows between two objects or systems due to a temperature difference between them is called heat.

• When energy stops being transferred (that is, when there is no net flow of heat between them), we say that the two objects are at the same temperature; they are in thermal equilibrium.

• Temperature is a scalar quantity that determines when objects are in thermal equilibrium.

• A thermometer measures temperature, but in order to do so it must be in thermal equilibrium with the object whose temperature it is measuring.



Zeroth Law of Thermodynamics

• If two objects are in thermal equilibrium with a third, then the two are in equilibrium with each other.

• If they are in equilibrium, they are at the same temperature.

• Without the zeroth law, it would be impossible to define temperature.


Temperature Scales

Lord Kelvin(1824 - 1907)

Anders Celsius(1701 - 1744)

Daniel G. Fahrenheit(1686 - 1736)


Water boils

Water freezes

212

32

Farenheit

100

0

Celcius

273.15

373.15

Kelvin

NOTE: K=0 is “absolute zero”, meaning (almost) zero KE/molecule

Temperature Scales

SI

373.15 K

Temperature Scales

• TC: temperature in Celsius scale– Unit: ˚C

• TF: temperature in Fahrenheit scale– Unit: ˚F

• TSI: temperature in SI unit– Unit: K (kelvin)

• Steam point: the boiling temperature of water at P = 1 atm– 212 ˚F = 100˚C = 373.15 K

• Ice point: the freezing temperature of water at P = 1 atm– 32 ˚F = 0˚C = 273.15 K

273.15 K



Temperatures


Sick ILQYou measure your body temperature with a thermometer calibrated in Kelvin. What do you hope the reading is (assuming you are not trying to fake some sort of illness) ? A) 307 K B) 310 K C) 313 K D) 317 K


Temp Scales ILQTwo cups of coffee are heated to 100 degrees Fahrenheit. Cup 1 is then heated an additional 20 degrees Centigrade, cup 2 is heated an additional 20 Kelvin. Which cup of coffee is hotter?

A) One B) Two C) Same

K = C + 273

ILQ• The temperature at which liquid nitrogen boils (at

atmospheric pressure) is 77 K.

What is this temperature in ˚C and ˚F, respectively?

a) 100˚C and 212˚Fb) -196˚C and –321˚Fc) 0˚C and 32˚Fd) –273.15˚C and –459.67˚Fe) –269˚C and –452˚F


ILQ

• Which of the following is the closest to 15˚C?

a) 8.3˚Fb) 27˚Fc) 40˚Fd) 60˚F


ILQ• A pot of water on the stove is heated from T1 =

25˚C to T2 = 100˚C. By what factor does the temperature in Kelvin change?

a) T2 = 4 ⋅ T1

b) T2 = 1.25 ⋅ T1

c) T2 = 0.80 ⋅ T1

d) T2 = 0.20 ⋅ T1



Internal Energy and Temperature• All objects have “internal energy” (measured in Joules)

– random motion of molecules• kinetic energy

– collisions of molecules gives rise to pressure• Amount of internal energy depends on

– temperature• related to average kinetic energy per molecule

– how many molecules• mass

– “specific heat”• related to how many different ways a molecule can move

– translation– rotation– vibration

• the more ways it can move, the higher the specific heat

Internal Energy• The internal energy of a system is the total energy of all of the

molecules in the system except for the macroscopic kinetic energy (kinetic energy associated with macroscopic translation or rotation) and the external potential energy (energy due to external interactions).

• Internal energy includes– Translational kinetic energy of the molecules

• the average translational kinetic energy of the molecules of an ideal gas <Ktr> = (3/2)⋅k⋅TSI (Note: TSI in K)

– Rotational and vibrational kinetic energy of the molecules– Potential energy between molecules– Chemical and nuclear binding energy of the molecules

• Internal energy does not include– any energy related to outside or macroscopic sources or motions, like

• overall translational energy of the system• potential energy due to external fields such as gravity.


Documents

Chapter 13: Temperature and the Ideal Gas