Physics 132 Review/Concept Summary
Last updated January 1, 2011
Note: Although I have done my best to check for typos and list the formulas correctly, you
should verify the formulas are correct before using them. Make sure you know what all the
variables represent in any particular formula. Some letters are used in different formulas from
different chapters and may represent different things.
- Dr. Nazareth
Newton’s Law of Universal Gravitation (13.3-13.4)
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•
•
•
•
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(magnitude of force)
m1 and m2 are mathematical points
G = 6.67×10-11 Nm2/kg2 = universal gravitation constant – same value for all pairs of
particles everywhere in the universe
It is an attractive force
The force that each particle exerts on the other is directed along the line joining the
particles
Also true for spherical bodies (like planets or stars), but distance is to center of the sphere
(planet)
Weight = gravitational force that the earth (or another astronomical body like the moon)
exerts on an object
o
(Weight = mg is the approximation)
o Weight always acts downward, toward the center of the earth
o An object has weight regardless of whether or not it is resting on the surface of
the earth
o Weight depends on G, the mass of the earth, ME, and distance, r, while the mass of
an object does not change
Kelpler’s Laws of Orbital Motion (13.6)
• First law: Planets follow elliptical orbits with the Sun at one focus of the ellipse.
• Second law: As a planet moves in its orbit, it sweeps out an equal amount of area in an
equal amount of time (See Figure 13-19 in textbook)
• Third law: The period, T, of a planet increases as its mean distance form the Sun, r,
raised to the 3/2 power. That is, T = (constant)r3/2
o For planet orbiting the Sun (circular orbit), T = (2π/sqrt(GMsun))r3/2
o Same formula works for orbiting the Earth or another planet – just replace the
mass of the Sun with the mass of the planet being orbitted
o Note the period depends on the mass of the object being orbited, not the mass of
the orbiting object
Satellites in Circular Orbits (13.6)
• Satellite is kept on circular path by gravitational force.
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•
(magnitude) →
If a satellite is to remain at radius, r, it must have speed, v, precisely.
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(magnitude)
•
Gravitational Potential Energy (13.5)
• PEg = -GmM/r
• PEg approaches zero as r approaches infinity
• Convenient to choose infinity as reference point for PEg = 0 when have distances on
astronomical scale
Energy Conservation (13.6)
• Mechanical energy of an object of mass, m, at distance, r, from the Earth
o ME = PE + KE = ½ mv2 – GmME/r
• Speed of asteroid striking earth, starting at infinite distance with v0 = 0
o vf = sqrt(2GME/RE) = 11,200 m/s ≈ 25000 miles per hour
• Escape speed of object just barely leaving Earth, with vf = 0 when reaches infinite
distance from Earth.
o This is the initial velocity it must have at the surface to escape Earth’s gravity
o v0 = sqrt(2GME/RE) = 11,200 m/s ≈ 25000 miles per hour
The Ideal Spring (Review)
Hooke’s Law (Restoring force of an ideal spring): F = -kx
- the minus sign means that the restoring force always points the opposite of the
direction of the displacement.
- k is the spring constant (units: N/m)
- x is the displacement of the spring from its unstrained length
Simple Harmonic Motion and Circular Motion (Reference Circle) (14.1-14.2)
We use the concept of the reference circle to derive formulas to describe the
displacement, velocity, and acceleration of an object undergoing simple harmonic
motion.
Period (the time for one cycle): T = 1/f = 2π/ ω
(units: s)
Frequency (number of cycles per second): f = 1/T
(units: Hz = 1/s)
Angular frequency: ω = 2πf = 2π/T
(units: rad/s)
(do NOT confuse the angular frequency, ω, of an object in simple harmonic motion
with the angular velocity, ω, of a body undergoing rotation or circular motion)
Amplitude: the object’s maximum displacement from equilibrium. Object oscillates
between x = -A and x = +A.
Displacement: x = A cos(ωt + Φ0)
Velocity: v = -Aω sin(ωt + Φ0)
- maximum velocity occurs at x = 0: vmax = Aω
Acceleration: a = -Aω2 cos(ωt + Φ0)
- maximum acceleration occurs at x = +A and x = -A: amax = Aω2
Phase: Φ = ωt + Φ0
- Φ0 = initial position of particle on reference circle at time t = 0
- “The phase is simply the angle of the circular motion particle whose shadow
matches the oscillator.” (pg 416, Physics, 2nd ed., by Randall Knight)
- the phase increases with time
Phase Constant: Φ0
- specifies initial conditions of the oscillator (at time t = 0)
x0 = AcosΦ0
v0x = -ωAsinΦ0
- different phase constants mean different initial conditions and different starting
locations on the reference circle
Frequency of vibration depends on mass and spring constant
Angular frequency:
or
Period:
Energy and Simple Harmonic Motion (14.3)
Elastic Potential Energy:
(SI units: J)
Total Mechanical Energy for a horizontal simple spring:
If there are no nonconservative forces (like friction), then mechanical energy is
conserved: E0 = Ef.
For a more complex system/object, we can write out the generic form of the total
mechanical energy formula to include elastic potential energy.
Dynamics of SHM (14.4)
- Fnet = Fspring_x = -kx = max
- Equation of motion for a mass of a spring:
- same solution as already derived with the reference circle
Vertical Spring (14.5)
If the ideal spring is vertical, the motion is still simple harmonic motion. The spring will
oscillate about an equilibrium position.
Equilibrium position:
Displacement: y = A cos(ωt + Φ0)
(The rest as before for the horizontal spring, just using y instead of x.)
Simple Pendulum (14.6)
For small angles only:
sin θ ≈ θ (θ in radians)
Angular frequency:
(for small angles)
Period:
(for small angles)
Conditions for Simple Harmonic Motion (14.6)
Need a linear restoring force when displaced from equilibrium position … then get SHM
Physical Pendulum (14.6)
Solid object that swings back and forth on a pivot under the influence of gravity
(I = moment of inertia of object)
Damped Oscillations (14.7)
-Drag force on a slowly moving object: D = -bv
-Damping constant, b, depends on shape of object and viscosity of air or other medium
damped oscillator
xmax(t) = Ae-bt/2m
Energy in damped systems (14.7)
- time constant, τ = m/b
(SI units = seconds)
- measures the time need for energy to decay to e-1, or ~37% of initial value
- xmax(t) = Ae-t/ 2τ
- mechanical energy at time, t: ½ k(xmax)2 = (½ kA2)e-t/τ = E0e-t/τ
- Approximately 2/3 energy gone after one time constant, 90% gone after 2τ
Driven Oscillations and Resonance (14.8)
- simple example is pushing a child on a swing
- natural frequency, f0: frequency that system oscillates at if left to itself
- e.g. natural frequency of a spring = sqrt(k/m)/2π
- driving frequency, fext: frequency of periodic external force applied to oscillating
system
- response (amplitude) of oscillation is larger the closer the driving frequency is to the
natural frequency of the oscillating system
- largest response (amplitude) when fext = f0. Called resonance.
Fluids and Elasticity (Chapter 15)
• Mass Density (15.1)
o ρ = m/V
Where m = mass and V = volume
• Pressure (15.2)
o P = F/A
o Where F = force of fluid acting perpendicular to surface and A = area of that
surface.
• Pressure and Depth in a Static Fluid (15.2)
o P2 = P1 + ρgh
 P2 is at the deeper point in the static fluid
 P1 is at the shallower point in the static fluid
o Pressure same at all points on horizontal line in connected static fluid
o Fluid rises to same height in all open regions of container
• Pascal’s Principle (15.2)
o Any change in pressure applied to a completely enclosed fluid is transmitted
undiminished to all parts of the fluid and enclosing walls.
o Hydraulic lift (15.3)
• Pressure Gauges (15.3)
o Gauge pressure = P2 - Patm = ρgh
 This formula describes the pressure gauge (measuring device), not in what
you are measuring (e.g., the gauge pressure at two different points in the
water pipes of a house)
 P2 = the absolute pressure (as in, not relative to atmospheric pressure)
• Archimedes’ Principle (15.4)
o Any fluid applies a buoyancy force to an object that is partially or completely
submerged in it; the magnitude of the buoyancy force equals the weight of the
fluid that the object displaces.
o FB = Wfluid displaced = ρf Vf g
o The volume of the fluid displaced is equal to the volume of the submerged part of
the object that is immersed in the fluid.
 Be careful to figure out how much of the volume is in the fluid (“below
water”) and how much is above the fluid (“above water”).
•
Equation of Continuity (15.5)
o Mass flow rate (ρAv) has the same value for every position along a tube that has a
single entry point and a single exit point for fluid flow.
o ρ1A1v1 = ρ2A2v2
•
 ρ = fluid density; A = cross-sectional area of tube; v = fluid speed
o Usually we assume an incompressible fluid (liquid), so ρ1 = ρ2 = ρ which gives us
 A1v1 = A2v2
(Equation of Continuity)
o Volume Flow Rate Q = vA
Bernoulli’s Principle (15.5) A statement of energy conservation
o P1 + ½ρv12 + ρgy1 = P2 + ½ρv22 + ρgy2
 Points 1 and 2 are two different locations in the “pipe” where the pressure
might be different, the fluid speed may be different, the cross-sectional
area may be different, and the height of the pipe might be different.
 We assume that the fluid is incompressible, so the density, ρ, does not
change. [Technically, we also assume the fluid is nonviscous, but we are
not covering viscosity in Phy 132.]
Elastic Deformation (15.6)
Stretching or Compression:
- Y = Young’s modulus (SI units: N/m2)
Shear Deformation:
- S = shear modulus (SI units: N/m2)
Volume Deformation:
Pressure,
(SI units = N/m2 = Pa)
- B = bulk modulus (SI units: N/m2)
- the minus sign means an increase in pressure leads to a decrease in volume
Stress and Strain (15.6)
Stress = force/area
Stress is proportional to strain
Strain: Compressional/Stretching
Shear
Volume
- please note that strain is a unitless quantity
(SI units: Pa = N/m2)
Waves (Chapter 20)
Two features common to all waves:
1) they are a traveling disturbance
2) waves carry energy from place to place
Two basic types of waves are transverse and longitudinal waves.
For periodic waves, we can define maximum amplitude, A, period, T, frequency, f = 1/T,
and wavelength, λ.
- Review figure 20.11 to see the maximum amplitude, A, period, T, and the
wavelength, λ, defined.
Speed of a wave:
(true for any wave - not wave pulse)
We can describe waves mathematically using the following formulas:
Wave moves toward +x direction:
Wave moves toward -x direction:
- In the above two equations, t and x are variables (where you are in time and
space), and A, f, ω, Φ0, k, and λ are set/defined for the particular wave being
described by the formula.
- k = wave number (analogous to ω) = 2π/λ
units = radians/m
- ω = angular frequency = 2π/T
units = radians/s
- Φ0 = phase constant (characterizes the initial conditions)
Speed of a Wave on a String (20.1)
Speed of a wave on a string:
(only applies to a wave on a string)
- where FT is the tension force in the string, and (µ = m/L) is the linear density (or
mass per unit length) of the string
Phase and Phase Difference (20.4)
Phase, Φ = (kx – ωt + Φ0)
Wave fronts in 2D and 3D waves are “surfaces of constant phase”
Phase difference, ΔΦ = Φ2 – Φ1 = (kx2 – ωt + Φ0) - (kx1 – ωt + Φ0)
ΔΦ = k(x2-x1) = kΔx = 2πΔx/λ
- Phase difference depends on ratio of separation Δx to wavelength.
- For adjacent wave fronts (λ apart), ΔΦ = 2π radians
Phase and phase difference will be important in chapter 21 for wave interference.
Speed of Sound, Sound Intensity, and Decibels (20.5, 20.6)
- Sound is a longitudinal pressure wave in air (usually in air, although sound also travels
thru liquids and solids)
- speed of sound in air at 20° C = 343 m/s
-depends on temperature (lower speed at lower temperatures)
Waves carry energy. Power = energy transported/sec. (SI units = J/s = W = watt)
Sound Intensity:
(SI units = W/m2)
- where P = power, and A = unit area the sound is distributed over.
- if the sound comes from a point source (power spread out equally in all
directions), then A = 4πr2 (the surface area of a sphere of radius r)
Intensity level in decibels:
(units = dB = decibels)
- where “log” means logarithm to the base 10 (NOT the natural logarithm), I is the
sound intensity in question, and I0 is the intensity of the reference level (usually
taken to be the threshold of hearing for humans, 1x10-12 W/m2).
- note that sound intensity, I (in W/m2) and intensity level, β (in dB) are not the
same thing, although they are related to each other.
- intensity level is given in a logarithmic scale.
- Make sure you understand how to go from I to β, and β to I.
- Review Appendix in the textbook if your math skills in exponents and logarithms
are a bit “rusty.”
Doppler Effect for Sound (20.7)
- the Doppler effect is the apparent change in frequency due to the motion of a sound
source (and/or observer)
General case (both source and observer moving):
- where fo is the frequency “heard” by the observer, fs is the frequency emitted by
the source, vo is the speed of the observer, vs is the speed of the source, and v is the
speed of sound.
- in the numerator, use the + sign if the observer is moving toward the source, and
use the – sign if the observer is moving away from the source
- in the denominator, use the – sign if the source is moving toward the observer,
and use the + sign if the source is moving away from the observer.
- if the observer is NOT moving, then vo = 0.
- if the source is NOT moving, then vs = 0.
Physical mechanism is different between source moving with observer stationary, and
source stationary with observer moving.
- Source moving: wavelength changes so different for observer
- Observer moving: observer intercepts a different number of wave fronts per second
so observes a different frequency than that put out by source
Light (20.5)
Light is an electromagnetic wave. It is a wave in the electromagnetic field.
Visible light is only one small part of the electromagnetic spectrum.
Largest wavelengths are radio waves. Smallest wavelengths are X-rays.
See Figure 20.22
Light travels fastest through a vacuum and slows down slightly in materials.
Speed of light in vacuum, c = 2.99,792,458 m/s = 3.00x108 m/s for our purposes
Index of refraction, n = speed of light in a vacuum/speed of light in material = c/v
nair = 1.0003 ≈ 1.00 for our purposes
“the frequency of wave is the frequency of the source. It does not change as the waves
moves from one medium to another.” Knight, 2nd edition, pg. 619. Since the speed
changes, but the frequency does not, the wavelength must also change in the new
medium.
λmaterial = λvacuum/n
Doppler Effect for Light (20.7)
Unlike sound, light does not require a medium to travel through, so we can’t compare
speeds of source or observer relative to a medium. We need Einstein’s theory of
relativity to get wavelength from moving source.
Light from receding source:
Red-shifted
Light from approaching source:
Blue-shifted
λ0 = wavelength emitted by source
vs = speed of source relative to observer
c = speed of light in a vacuum
Red-shifted does not mean becomes red, it just means it shifts toward the longer
wavelength (red end) of visible light spectrum relative to the source wavelength
Blue-shifted does not mean becomes blue, it just means it shifts toward the shorter
wavelength (blue end) of visible light spectrum relative to the source wavlength
Linear Superposition of Waves (21.1)
- superposition occurs when two (or more) waves are traveling thru the same space at the
same time. The resultant disturbance is the sum of the two (or more) waves.
Transverse and Longitudinal Standing Waves (21.2-21.4)
- another case of interference can occur when we have waves reflected back and forth –
two waves traveling in the opposite direction, but over the same space
- only certain conditions will produce certain standing waves patterns.
- for standing waves on a string, this condition is related to the speed of waves on a
string.
(section 20.1, wave on a string)
- called “standing wave” because crests and troughs “stand in place”
- nodes: points in standing wave that never move
String: particle of string does NOT move up and down
(Longitudinal) sound wave: air molecule does NOT move back and forth
(Note: air pressure is at this point is at a maximum relative to normal air
pressure)
- nodes are places of destructive interference
- antinodes: points in standing wave with maximum displacement
String: particle of string moves up and down with maximum amplitude
(Longitudinal) sound wave: air molecule moves back and forth with max. amp.
(Note: air pressure at this point is at a minimum relative to normal air
pressure)
- antinodes are places of constructive interference
Mathematics of standing waves:
Add left traveling and right traveling sinusoidal waves together
(Remember angular frequency, ω = 2πf , and wave number, k = 2π/λ)
D(x,t) = DR + DL = asin(kx – ωt) + asin(kx + ωt)
Use trigonometric identity …
D(x,t) = DR + DL = (2asinkx) cosωt = A(x) cosωt
Amplitude function, A(x) = 2asinkx
Maximum of 2a where sin kx = 1
Transverse standing waves (string fixed at both ends):
(m = 1, 2, 3, …)
-m = 1 is the fundamental frequency, m = 2 is the 2nd harmonic (f2 = 2 f1), m = 3 is
3rd harmonic (f3 = 3 f1), etc
- The possible standing waves on a string of length, L, are called the normal
modes of the string and correspond to the value of m. m is the number of
antinodes on the string (NOT nodes)
Longitudinal standing waves - resonance in air column (tube open/closed both ends):
(m = 1, 2, 3, …)
Longitudinal standing waves (tube open one end and closed at other end):
(modd = 1, 3, 5, …)
Constructive and Destructive Interference of Waves (21.5-21.7)
- maximum constructive interference occurs when two waves are exactly in phase (crests
line up)
- perfect destructive interference occurs when two waves are exactly out of phase (the
crests from one wave line up with the wave troughs from the second wave)
- If you have two wave sources vibrating in phase, then you can get constructive or
destructive interference at different locations depending on the difference in path length
from the two sources, to the location in question.
Maximum constructive interference: ΔΦ = 2πΔx/λ + ΔΦ0 = m 2π rad
For identical sources, ΔΦ0 =0 rad, so Δx = mλ
Perfect destructive interference: ΔΦ = 2πΔx/λ + ΔΦ0 = (m+ ½) 2π rad
For identical sources, ΔΦ0 =0 rad, so Δx = (m + ½) λ
m = 0, 1, 2, …
m = 0, 1, 2, …
Thin Film Interference (21.6)
Index of Refraction:
(n ≥ 1)
Wavelength of light in a material:
Because light travels at different velocities in different materials, light “bends” or refracts
at the interface between two materials.
The index of refraction of a material depends slightly on the wavelength of light. This
leads to dispersion – the spreading of light into its color components. Examples of
dispersion of light: prisms and rainbows.
This is another case of interference. When light falls on a thin film, some of the light is
reflected immediately, while some refracts into the thin film, reflects off the interface at
the “bottom” of the thin film, and refracts back, producing a light ray that is parallel to
the original reflected ray. These “two” rays can interfere either constructively or
destructively, depending on the situation. The interference depends on two things: 1)
the extra path length that ray 2 (the one that went into the thin film before coming back
out) and 2) any possible phase change (“flip”) that might occur during reflection of the
rays.
Path length difference for ray 2 = 2d (two times the thickness of the thin film)
Reflection phase changes: depends on the situation.
- If ray 1 undergoes a phase change (“flip”) but ray 2 does not, then the contribution
is ½ λfilm
- If ray 2 undergoes a phase change (“flip”) but ray 1 does not, then the contribution
is ½ λfilm
- If both rays undergo a phase change (“flip”), then the contribution is nothing
because both rays have the same orientation.
How to tell if a phase change (“flip”) occurs for a reflection
- If light is going from n1 → n2, and n1 < n2, then there will be a phase change (“flip”)
- If light is going from n1 → n2, and n1 > n2, then there will NOT be a phase change
Another thing to remember is that the wavelength that matters in the interference
relationship is the wavelength within the thin film:
Review the class lecture notes and the examples in the textbook to see an interference
relationship worked out. (There is no one formula for me to write in the notes because it
depends on constructive/destructive interference, the thickness of the thin film, and
which phase changes occur for the reflections.)
Constructive and Destructive Interference of Waves in 2D and 3D (21.7)
- If you have two wave sources vibrating in phase, then you can get constructive or
destructive interference at different locations depending on the difference in path length
from the two sources, to the location in question.
In 2D or 3D, a wave moves out from source as a circular (2D) or spherical wave (3D)
D(r,t) = asin(kr – ωt + Φ0)
r = distance measured outward from source
Maximum constructive interference: ΔΦ = 2πΔr/λ + ΔΦ0 = m 2π rad
For identical sources, ΔΦ0 =0 rad, so Δr = mλ
Perfect destructive interference: ΔΦ = 2πΔr/λ + ΔΦ0 = (m+ ½) 2π rad
For identical sources, ΔΦ0 =0 rad, so Δr = (m + ½) λ
m = 0, 1, 2, …
m = 0, 1, 2, …
You get a more complicated pattern of antinodal lines (lines of points with maximum
constructive interference) and nodal lines (lines of points with perfect destructive
interference). See textbook for figures.
Beats (21.8)
- beats occur when you add two waves with slightly different frequencies.
- inference of waves varies the amplitude in a periodic fashion = modulation
Beat frequency:
fbeat = |f1 – f2|
Macroscopic Description of Matter (Ch 16)
Bulk properties – properties of system as a whole
- mass, volume, density, temperature, pressure
Phases of matter – solid, liquid, gas
- solid: rigid; definite shape and volume; atoms vibrate about equilibrium
positions but are not free to move inside solid; nearly incompressible
- liquid: fluid; liquid flows to fit shape of container; molecules are free to move
around inside liquid but are loosely held together by weak molecular bonds;
nearly incompressible
- gas: every molecule moves through space freely until collides with another
particle or wall of container; highly compressible; fluid
Phase change – change between solid and liquid, or liquid and gas
State variables – used to characterize a system; describe state of system; not all
independent of each other
examples of state variables: volume, pressure, mass, mass density, thermal
energy, moles, number density, temperature
Δ means change in a particular variable and always means final - initial
Thermal equilibrium – state variables are constant and not changing
Atoms and Moles (16.2)
Number density – number of atoms or molecules per cubic meter in a system; uniform
whether you look at whole or just part of system
= N/V
(SI units = m-3)
Atomic mass number, A = number of protons + number of neutrons in an atom of a
particular element; atomic mass number is different from atomic number (=
number protons)
Atomic mass scale defined using Carbon-12 (A = 12); 12C = 12 u
Atomic mass unit, u; 1 u = 1.66 x 10-27 kg
Molecular mass: sum of atomic masses of atoms composing molecule
Mole, mol : amount of substance containing same number of particles as atoms in 12 g
of 12C
Monatomic gas: basic particle composed of one atom
Diatomic gas: basic particle composed of two atoms
NA = Avogadro’s number of particles = 6.022x1023 particles in one mol; SI units = mol-1
Number of moles of a substance, n = N/NA
mp = mass of a single particle
ms = mass of sample (in grams)
Mmol = mass per mole of the gas (in grams)
Molar mass, Mmol = mass in grams of 1 mol of the substance
Temperature Scales (16.3)
Absolute zero: temperature at which all molecular motion would cease; the pressure
would be zero because pressure is caused by collisions of molecules; thermal
energy of system would be zero
Absolute temperature scale: temperature scale with zero point at absolute zero
Kelvin scale is a absolute temperature scale
Temperature in Kelvins (K): T = Tc + 273.15
(units = “Kelvins”, not “degrees Kelvin”)
Know which type of temperature scale you need to use for the problem. Look at the
units in equations as a check for whether you need to use Celsius degrees or Kelvins.
Thermal Expansion
This is not covered by your textbook but would normally be part of
the Phy 132 topics that another institution would expect you to have covered. Linear thermal
expansion was covered in you Phy 132 lab class.
Linear Thermal Expansion:
α = coefficient of linear expansion; units = 1/Cº = (Cº)-1)
Use this formula for one-dimensional expansion (or contraction) problems. The onedimension could be length, or it could be radius.
Volume Thermal Expansion:
β = coefficient of volume expansion; units = 1/Cº = (Cº)-1)
Phase Changes (16.4)
Melting point: temperature at which solid becomes a liquid
Freezing point: temperature at which a liquid becomes a solid
Phase equilibrium: point at which any amount of two different phases can exist at the
same time; line between phases on a phase diagram
Condensation point: temperature at which a gas becomes a liquid
Boiling point: temperature at which a liquid becomes a gas
Phase diagram: graph of pressure versus temperature of a particular substance used to
show how the phases and phase changes vary with pressure and temperature
See figure 16.4 (pg. 488) for example of phase diagram
Sublimation: when a solid goes straight to the gas phase without going through liquid
phase first
Critical point: end of the liquid gas boundary on a phase diagram where the substance is
a fluid but there is no distinction between gas and liquid at pressures and
temperatures above this point
Triple point: the single point on phase diagram where phase boundaries meet; can have
gas, liquid, and solid in equilibrium at this point; reference point for the Kelvin
scale; for water triple point is at 0.01°C and 0.006 atm.
Ideal Gas Law (16.5)
Ideal gas law:
n = number of moles of gas
R = universal gas constant = 8.31 J/(mol K)
T = temperature in Kelvins
Alternative version:
N = number of gas particles
k = Boltzmann’s constant = 1.38x10-23 J/(K molecules)
Boyle’s Law (constant T, constant n):
Charles’ Law (constant P, constant n):
Ideal-Gas Processes (16.6)
Ideal gas process: “means by which the gas changes from one state to another.” (pg.
494, Physics, 2nd ed., Knight)
PV diagram: diagram of pressure versus volume (pressure on “y”-axis, volume on “x”)
Assume number of moles, n, is constant
Each point on graph represents a unique state of graph (use PV and n to get T)
Quasi-static process: process occurs slowly enough that P, V and T are essentially same
as equilibrium values and same throughout gas; an idealization, but a good
approximation in many real life cases; reversible process
Thermal Processes (16.6 & 17.2 & 17.4)
Isobaric (constant pressure)
P0 = Pf
Example: gas in cylinder with piston with constant amount of weight on top; piston
keeps atoms from escaping, but can freely move up and down to change volume
Note: constant weight of piston means constant pressure!
P = Patm + Mg/A
Horizontal line on PV diagram
W = -PΔV = -P(Vf – Vi)
Work = -area under the curve on a P vs V diagram.
Isochoric = Isovolumetric (constant volume)
V0 = Vf
Example: gas in closed, very rigid container that doesn’t change shape as pressure
changes
Vertical line on PV diagram
W = 0 = no work done
ΔEth = Q
Isothermal (constant temperature)
T0 = Tf
P0V0 = PfVf
Example: gas in cylinder in constant temperature surroundings so that heat transfer
through walls of container keep gas at same temperature as surroundings
Hyperbolic curve on PV diagram
Isotherm: hyperbolic curve on PV diagram representing a particular temperature
(for an ideal gas)
ΔEth =0 (no temperature change)
Q = -W
It’s all about energy (17.1)
Thermal energy: energy in motion of particles and spring like molecular bonds
Eth = KEmicro + PEmicro = Eint
Note: ignoring chemical, nuclear, and other forms of internal energy (more advanced
course)
Total energy of system: ΔEsys = ΔEmech + ΔEth = Wext + Q
Work in Ideal-Gas Processes (17.2)
Work: energy transferred between system and environment through mechanical
interaction; net force acts over a distance
Signs: positive when energy transferred from environment into system
negative when energy transferred from system to environment
W = - area under the curve on a PV diagram between V0 and Vf
W > 0 when gas compressed (environment doing work on the gas)
Gas gains energy from the environment
W < 0 when gas expands (gas doing work on the environment)
Gas loses energy to the environment
W = 0 if volume doesn’t change
First Law of Thermodynamics (17.4)
ΔEth = W + Q
Q = heat; W = work; ΔEth = (Eth)f – (Eth)0 = change in internal thermal energy of a system
This is just energy conservation.
Make sure you know the sign conventions for Q and W.
Heat, Temperature Change, and Phase Change (17.3, 17.5-17.6)
Heat is energy transferred between the environment and a system because there is a
temperature difference between them
SI units for heat (energy) = joule (J)
(= 1 Nm)
1 calorie = 1 cal = quantity of heat needed to change temperature of 1 g of water by 1°C
= 4.186 J
1 food calorie = 1 Cal = 1000 cal = 1 kcal = 4186J
(Food calorie is what is listed under the nutrition label on the food box)
Heat flows from hotter object to colder object through energy transfer during molecular
collisions
Thermal equilibrium is when the environment and system or two systems are at the
same temperature and there is no heat flow.
Temperature and heat are NOT the same thing. Make sure you understand the
difference.
Thermal energy is energy of system due to motion of its molecules.
Heat is energy transferred due to a temperature difference.
Temperature is a state variable that describes how “hot” or “cold” the system is.
Heat supplied (or removed) in changing the temperature of a substance:
c = specific heat capacity; units = J/(kg K)
Heat capacity of a material in terms of molar specific heat capacity, C:
Note that C is in units of J/(mol·K), and ΔT is in units of K.
Heat supplied (or removed) in changing the phase of a substance:
Q = CnΔT
L = latent heat of the substance; units = J/kg
You must add sign to indicate adding heat or removing heat to cause a phase change.
Use the latent heat of fusion, Lf, for the change between solid and liquid phases.
Use the latent heat of vaporization, Lv, for the change between liquid and gas (vapor)
phases.
Use the latent heat of sublimation, Ls, for the change between solid and gas phases.
Calorimetry: Qnet = Q1 + Q2 + Q3 + … = 0 = QΔT + Qphase
Signs are VERY important!
ΔT = Tf – T0
always! So sign comes automatically for Q = cm ΔT
You must supply proper sign for phase changes: Q = ±mL
Melting and vaporizing requires energy to enter substance
Freezing and condensing requires energy to leave substance
Remember that you might first have to change the temperature of a substance to the
temperature where a phase change could occur, then change the phase, then change the
temperature to the “final” temperature. You would need some amount of heat for each.
Specific Heat of Gases (17.7)
Q = nCVΔT (temperature change of gas at constant volume)
Q = nCPΔT (temperature change of gas at constant pressure)
CP = CV + R
ΔEth = nCVΔT
(for any ideal-gas process!)
Adiabatic expansion or compression (no heat flow into or out of system)
Q =0
W = nCVΔT
where
Convection (17.8)
Heat is transferred by bulk movement of a gas or liquid.
Conduction (17.8)
Heat is transferred directly through a material without any bulk motion of the substance.
Conduction of Heat thru a Material:
k = thermal conductivity of the material; units = J/(sm K)
ΔT = the change in temperature from one end of the material to the other.
We treat the heat conduction as steady state. This means that the amount of heat per unit
time is constant through each material (when you have several materials sandwiched
together). If the rate of heat transfer by conduction was not constant, then we would
have heat accumulating (or disappearing).
Radiation (17.8)
Heat is transferred by means of electromagnetic waves. This mode of heat transfer does
not require a material to transfer energy.
Stefan-Boltzmann Law of Radiation:
e = emissivity; 0 ≤ e ≤ 1; e = 1 for a perfect emitter; units = none
σ = Stefan-Boltzmann constant = 5.67x10-8 J/(sm2K4)
Objects can both emit and absorb radiation simultaneously. If an object has a higher
temperature than its surroundings, the object emits a net radiant power:
T = temperature of the object
T0 = temperature of the surroundings
Molecular Speed and Collisions (18.1)
- molecules in a substance have a distribution of speeds
Mean free path = average distance of particle between collisions
Pressure in a gas (18.2)
Root mean square speed, vrms = sqrt((vx2)avg + (vy2)avg + (vz2)avg)
Temperature (18.3)
Average translational kinetic energy of a molecule:
Temperature measures the average translational kinetic energy of a gas
T in Kelvins!
T in Kelvins!
Thermal energy and specific heat (18.4)
Monatomic means gas molecules are composed of single atom particles
Thermal Energy of a Monatomic Ideal Gas:
Constant volume specific heat for a monatomic gas: CV = (3/2)R
Thermal Energy of a Solid: Eth = 3NkT = 3nRT
Thermal energy of a diatomic ideal gas: Eth = (5/2)NkT = (5/2)nRT
Constant volume specific heat for a diatomic gas: CV = (5/2)R
Thermal equilibrium (18.5)
“heat is the energy transferred via collisions between the more energetic (warmer) atoms
on one side and the less energentic (cooler) atoms on the other.” (Pg 555, Knight,
Physics for Scientists and Engineers, 2nd edition)
At thermal equilibrium (ε1)avg = (ε2)avg
and
T1f = T2f = Tf
Second Law of Thermodynamics (18.6)
Irreversible process: a process that can only happen in on direction
Entropy: the “measure that a macroscopic state will occur spontaneously.” (Pg 558
Knight, Physics for Scientists and Engineers, 2nd edition)
Second law of thermodynamics: “The entropy of an isolated system (or group of
systems) never decreases. The entropy either increases, until the system reaches
equilibrium, or, if the system began in equilibrium, stays the same.” (Pg 559 Knight,
Physics for Scientists and Engineers, 2nd edition)
An informal statement of the 2nd law is that heat flows spontaneously from the hotter
system to the colder system, but never the opposite way.
Heat Engines (Ch 19)
In this chapter, instead of looking at the work done on the system as we did in chapter 17,
we want to consider the work done by the system.
Work done by the system = negative work done on the system
Ws = -W = (positive) the area under the PV curve
A heat engine is any device that uses heat to perform work.
Three essential features: 1) input heat, QH, from hot reservoir; 2) part of heat used to
perform work, W, by the engine; 3) rest of heat (output heat, QC) rejected to cold
reservoir.
An energy reservoir is an object or part of the environment so vast that its temperature
doesn’t change when heat is added or removed by the heat engine.
Closed cycle device: returns to its initial conditions periodically
If no other losses in engine: Wout = QH - QC
(work done per cycle)
Thermal efficiency of engine:
= what you get/what you had to pay
(η ≤ 1)
ηperfect = 1 (but this is NOT possible due to 2nd law of thermodynamics)
usual numbers for efficiency = 0.1-0.5
Some waste heat must be exhausted into the cold reservoir.
A refrigerator or air conditioner is a closed cycle device that uses work to remove heat
from a cold reservoir and put it into the hot reservoir. It has to put more heat into the hot
reservoir (outside the refrigerator) than it removes from the cold reservoir (inside the
refrigerator)
Coefficient of performance, K= QC/Win = what you get/what you had to pay
Kperfect = ∞ (but NOT possible due to 2nd law of thermodynamics)
Ideal gas law heat engines use ideal gas as the working substance of the heat engine.
Wout = Wexpand-|Wcompress| = area inside the closed curve on PV diagram (one full cycle)
*** Table 19.1 summarizes results for specific ideal gas processes !!!!
A Carnot engine maximizes efficiency.
TC, and TH must be in Kelvins.
Efficiency of Carnot engine:
Carnot coefficient of performance: KCarnot = TC/(TH-TC)