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Study Guide For Final Exam
Physics 2210 – Albretsen
Updated: 06/09/2017
All Other Previous Study Guides – Modules 01 - 15
Module 16
Sound Waves
•
Speed of sounds waves depends on the compressibility and inertia of the medium
v=

elastic property
inertial property
β
v= ρ
√
β = Bulk Modulus
ρ = Equilibrium Density
v=
Y = Young's Modulus
ρ = Equilibrium Density
Intensity
I=
•

Y

Average Power
Area
Sound levels in Decibels (dB)
 = 10 log
 
I
I0
I0 = Reference Intensity = 1.00 × 10-12 W / m2 – Threshold of hearing
β = Sound Level
Standing Waves in Air Columns
•
L = Pipe Length
Open / Open Pipe:
• Both ends of pipe will have an antinode
v
f n=n
n =1, 2, 3, 4, ...
2L
Open / Closed Pipe
• Open end will have an antinode, closed end will have a node
f n=n
v
4L
n = 1, 3, 5, 7, ...
Beats
•
•
•
•
•
Superposition of two waves with slightly different frequencies traveling in the same
direction.
Periodically goes in and out of phase.
Beat – periodic variation in the intensity at a given point due to the superposition of two
waves having slightly different frequencies (f1 and f2).
Beat frequency (fb)– number of beats heard per second – human ear can detect about
20 beats / second.
f b=| f1− f2 |
Resultant vibration:
f1 f2
2
Phase Difference
•
Phase difference between to points on a sinusoidal wave:
x
 =2 

Doppler Effect
•
Apparent change in frequency experienced by an observer whenever there is relative
motion between the source and the observer.
f = Source frequency
f' = Observed frequency
v0 = Velocity of observer
vS = Velocity of source
v = Velocity of wave in media (air, water, etc etc)
Observer moving toward source:

v0
v


v0
v

f '=f 1
Observer moving away from source:
f '=f 1−
Source moving toward observer:
f '=f
 
1
1−
vs
v
Source moving away from observer:
f '=f
 
1
1
vs
v
Source and observer in motion:
f '=f
 
v ± v0
v ∓ vs
Module 17
Temperature
•
•
•
•
Our human perception of temperature can be misleading.
Thermal Contact – Two objects are in thermal contact if heat can be exchanged
between them.
Thermal Equilibrium – When two objects in thermal contact cease to have any
exchange of heat.
Examples of different types of thermometers and issues with each.
Zeroth Law of Thermodynamics – If objects A and B are separately in thermal
equilibrium with a third object C, then A and B are in thermal equilibrium with each
other if placed in thermal contact.
Temperature Scales
1.
2.
3.
4.
Kelvin (T) – Absolute Scale
Celsius (TC)
Fahrenheit (TF)
Rankine (TR) – Absolute Scale
T C = T − 273.15
9
T F = T C  32
5
5
 T = TC =  TF
9
 TF = TR
Thermal Expansion of Solids and Liquids
•
•
Object has initial length L0
Increases length ΔL when ΔT is small
L − L0 =  L =  L0  T
 = Average coefficient of linear expansion for a given material
•
Different materials:
1.
2.
3.
4.
5.
Expand only one direction
Or expand in one way and contract in another
Or contract
Or expand in all three dimensions
Or expand in only two dimensions
V − V 0 =  V =  V 0  T = 3  V 0  T (Assumes equal expansion in all three directions)
 = Average coefficient of volume expansion for a given material
A − A 0 =  A = 2  A0  T
Specific Heat
•
•
When heat is added to a substance, the temperature rises except during a phase
change.
Heat Capacity (C') - depends the amount of heat (thermal energy, Q) needed to raise a
sample one degree Celsius – depends on substance.
Q=C '  T
•
Specific Heat ( c ) – Heat capacity per unit mass – depends on substance - look up in
tables.
Q = cm  T
Latent Heat
•
Thermal energy is transferred between substance and surroundings with no change in
internal temperature – Phase change.
Q = Δ mL
L = Latent heat of a particular substance
L f = Latent heat of fusion (solid to liquid)
Lv = Latent heat of vaporization (liquid to gas)
Types of Heat Transfer
•
Conduction – Energy flows within a body or between two bodies in thermal contact
•
Convection – Transfer of heat by motion of mass (fluid) from one region to another
•
Radiation – Transferred by electromagnetic radiation (light)
Module 18
Ideal Gas Law
•
•
Assume very low pressure (measured in Pascals) and low density
Express amount of gas in number of moles (n)
m = Mass of substance
M = Molar mass of substance
n=
•
m
M
Express Temperature in Kelvin and Volume in cubic meters
PV = nRT
R = Universal Gas Constant = 8.31
•
J
mol K
Can also express in total number of molecules (N)
N A = Avogadro's Number = 6.022 × 1023
N = nN A
PV = Nk B T
R
J
kB =
= 1.38 × 10−23
= Boltzmann's Constant
NA
K
PV Diagrams
•
•
Graph of pressure as a function of volume
Curve for a specific temperature is called an isotherm
Kinetic Theory Of Gasses
•
Temperature is a direct measure of the average molecular kinetic energy.
1
3
m v¯2 = k B T
2
2
◦ The number of molecules is large and the average separation between molecules
is large compared to their size.
◦ The molecules obey Newton's Laws of Motion, but as a whole they move randomly.
◦ The molecules undergo elastic collisions with each other.
◦ Collisions with the walls are elastic on average.
▪ Kinetic energy and momentum are conserved!
◦ The gas is a pure substance.
•
Equipartition of energy:
◦ Each translational degree of freedom contributes equal amounts of energy
◦ The total translation of all kinetic energy:
E=N
•
( 12 m v¯ ) = 32 N k
2
B
T=
3
nRT
2
The square root of average value of the square of the velocity is called the root mean
square (rms) speed of the molecules:
√
√
3 kB T
3 RT
v rms = √ v¯2 =
=
m
M
Collisions Between Molecules
•
A more realistic model has molecules colliding. How often do they collide?
•
Simple model:
◦ N spherical molecules in volume V
◦ r = radius of a molecule
◦ Two molecules collide – The distance between their centers is 2r
◦ Assume only one molecule is moving at velocity v
◦ Imagine the moving molecule traveling down a center of a cylinder with radius 2r
◦ Any stationary molecule whose center is inside the cylinder will collide with the
moving molecules
V
t mean =
Average time between collisions, or the mean free time:
4 π √ 2 r 2 vN
•
•
(
)
The average distance traveled is just the mean free time multiplied by the velocity of
the molecule, otherwise known as the mean free path.
λ=
(
V
4 π √2 r 2 N
)
Module 19
Heat and Thermal Energy
•
•
•
•
Thermal energy is part of the internal energy of a stationary system
Thermal energy transfer caused by a temperature difference between system and
surroundings
Work is done “on” or “by” a system – measure of energy transfer between the system
and surroundings
• Work done “of” a system makes no sense
Energy can be transferred between two systems even when no thermal energy transfer
occurs in the form of work
Work and Thermodynamic Processes
•
Depends on the path taken on PV diagram (Area under the curve)
Vf
W = ∫V P dV
i
First Law of Thermodynamics
“You can't get something for nothing.”
•
Energy can be transferred between a system and its surrounds in two ways:
1. Work done by or on the system (W)
• Positive W – Work done by the system on surroundings (leaves system)
• Negative W – Work done on the system by surroundings (enters system)
2. Thermal energy transfer (Q)
• Positive Q – Heat / Thermal energy is transferred into the system from
surroundings
• Negative Q – Heat / Thermal energy leaves the system
•
•
•
Q and W are dependent on path taken on PV diagram
Q – W are independent of path taken on PV diagram
U = Internal energy
Δ U = U f − Ui = Q − W
•
Special Cases:
1. Isolated system – system does not interact with surroundings – internal energy is
constant
Q =0
W =0
Δ U =0
2. Cyclic Process – Internal energy originates and ends in the same state
Uf =Ui
Δ U =0
Q=W
Three Basic Gas Processes
•
Adiabatic Process – No thermal energy enters or leaves the system
◦ Perform process rapidly so heat has no time to flow
Q= 0
Δ U =−W
Vf
•
•
Isochoric Process (Constant Volume):
W = ∫V P dV = 0
Isobaric Process (Constant Pressure):
W = ∫V P dV = P (V f − V i )
i
 U =Q
Vf
i
Q = Non-zero
•
Isothermal Process (Constant Temperature):
• Assume ideal gas
Vf
Vf
W = ∫V P dV =
i
∫V
nRT
i
V
W = nRT ln
Vf
dV = nRT ∫V
i
dV
V
 
Vf
Vi
Module 20
Directions of The Thermodynamic Processes
•
•
•
Natural processes are irreversible processes
◦ Increases randomness / disorder of the system
Naturally proceeds in one direction, but not the other
Heat flows form a hotter body to a cooler body, but never in reverse
Heat Engines
•
•
Device that takes heat from a source and converts it to mechanical work
Working substance – mass (gas, liquid, solid) that undergoes inflow and outflow of
heat, expansion, and compression
•
Hot Reservoir – Heat source that can give the working substance large amounts of
heat at a constant temperature without appreciably changing its own temperature
Cold Reservoir – Can absorb large amounts of discarded heat from the engine at a
constant lower temperature
•
•
The net heat absorbed per cycle is:
◦ QH – Positive
Q = QH + QC = |Q H| − |QC|
◦ QC - Negative
•
The work done is:
•
In a magical real world, we'd like to convert all of QH into Q and have nothing for QC.
◦ Experience shows this can never be the case.
•
Thermal efficiency of and engine:
W = Q = QH + QC = |Q H| − |Q C|
e=
QC + Q H
QC
QC
W
=
=1+
=1−
QH
QH
QH
QH
| |
Refrigerators
•
•
Heat engine operating in reverse
Mechanical work takes heat from the cold reservoir and places in in a hot reservoir
◦ QC is positive
−Q H = QC − W
◦ QH and W are negative
•
A coefficient of performance can be defined:
|Q |
|QC|
K= C =
|W | |Q H| − |QC|
The Second Law of Thermal Dynamics
“You can't even break even.”
“It is impossible for any system to undergo a process in which it absorbs heat from a
heat reservoir at a single temperature and converts the heat completely into
mechanical work, with the system ending in the same state in which it began.”
“It is impossible for any process to have as its sole result the transfer of heat from a
cooler to a hotter body.”
Carnot Cycle
•
•
What is the most efficient you can get?
Idealized heat engine called the Carnot Cycle has the maximum possible efficiency
and still consistent with the Second Law of Thermodynamics.
•
For maximum efficiency of a heat engine, we must avoid all irreversible processes:
◦ For this to happen, the temperature of the hot reservoir must be the same as the
working substance (TH).
◦ Also, the temperature of the working substance must be the same as the cold
reservoir when heat is discarded into it (TC).
◦ Any finite temperature drop would result in an irreversible processes.
◦ Every process that involves heat transfer must be isothermal.
◦ Any process in which the the working substance is between TH and TC, there must
be no heat transfer into the hot or cold reservoirs, it must be an adiabatic process.
Steps of the Carnot Cycle
1.
2.
3.
4.
Gas expands isothermally at temperature TH, absorbing heat QH
Gas expands adiabatically until its temperature drops to TC
Gas compressed isothermally at TC, rejecting heat QC
Gas compressed adiabatically back to its originally temperature TH
◦ The efficiency of of a Carnot Engine will be:
e Carnot = 1 −
TC TH − TC
=
TH
TH
▪ When the temperature difference is large, the efficiency is large
▪ The efficiency can never be unity unless TC = 0
◦ Similarly, the coefficient of performance for a Carnot Refrigerators
▪ Each step is reversible, so the process can go in reverse:
TC
K Carnot = 1 −
T H − TC
“No engine can be more efficient than a Carnot Engine operating between the two
same temperatures.”
“All Carnot engines operating between the same two temperatures have the same
efficiency, irrespective of the nature of the working substance.”
The Kelvin Temperature Scale
•
•
The ratio of QC / QH must be the same for all Carnot engines between TH and TC
Kelvin proposed we define ratios of heat absorbed and temperatures to be the
same:
T C |Q C|
Q
=
=− C
T H |QH|
QH
•
This defines a temperature scale based on the Second Law of Thermodynamics
instead of an ideal gas
◦ Independent of any particular substance
◦ Thus the Kelvin scale is truly absolute
•
•
•
Theoretically absolute zero cannot be attained experimentally
The more closely we approach absolute zero, the more difficult it is to get closer
The Third Law of Thermodynamics:
“You can't stop playing the game”
It is impossible to reach absolute zero in a finite number of thermodynamic steps
Entropy
•
•
The Second Law of Thermodynamics can be stated as a quantitative relationship with
the concept of entropy
Irreversible heat flow increases randomness (remember Zeroth Law)
“Isolated systems tend toward disorder, and entropy is a measure of this disorder”
•
A reversible process:
◦ If the total amount of heat Q is added during a reversible isothermal process, the
change in entropy is:
Q
Δ S = S2 − S 1 =
T
◦ As infinitesimal amounts of heat are added to a reversible process, integrate:
final state
Δ S = ∫initial state
dQ
T
•
Entropy is the measure of the randomness of a system specific state
◦ Only depends upon current state of the system
◦ Does not depend on past history
•
When going from an initial state to a final state, the change in entropy does not depend
upon the path taken
◦ The change in entropy will be the same for all possible paths from initial to final
state
•
To find the change in entropy for an irreversible process, simply invent a path between
the initial and final states that consists of only reversible processes
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