Physics 122 Review



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)

• [pic] (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 [pic] (Weight = mg is the approximation)[pic]

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.

• [pic] (magnitude) → [pic]

• If a satellite is to remain at radius, r, it must have speed, v, precisely.

• [pic] (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: [pic] or Period: [pic]

Energy and Simple Harmonic Motion (14.3)

Elastic Potential Energy: [pic] (SI units: J)

Total Mechanical Energy for a horizontal simple spring:

[pic]

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.

[pic]

Dynamics of SHM (14.4)

- Fnet = Fspring_x = -kx = max

- Equation of motion for a mass of a spring: [pic]

- 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: [pic]

Displacement: y = A cos(ωt + Φ0)

(The rest as before for the horizontal spring, just using y instead of x.)

Simple Pendulum (14.6)

[pic]

For small angles only:

sin θ ≈ θ (θ in radians)

Angular frequency: [pic] (for small angles)

Period: [pic] (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

[pic] (I = moment of inertia of object)

[pic]

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

[pic]

[pic] damped oscillator

[pic]

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 ( = 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 + (gh

▪ 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 = (gh

▪ 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 ((Av) 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 (1A1v1 = (2A2v2

▪ ( = fluid density; A = cross-sectional area of tube; v = fluid speed

o Usually we assume an incompressible fluid (liquid), so (1 = (2 = ( 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 + ½(v12 + (gy1 = P2 + ½(v22 + (gy2

▪ 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, (, 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: [pic]

- Y = Young’s modulus (SI units: N/m2)

Shear Deformation: [pic]

- S = shear modulus (SI units: N/m2)

Volume Deformation:

Pressure, [pic] (SI units = N/m2 = Pa)

[pic]

- 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 (SI units: Pa = N/m2)

Stress is proportional to strain

Strain: Compressional/Stretching [pic]

Shear [pic]

Volume [pic]

- please note that strain is a unitless quantity

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: [pic] (true for any wave - not wave pulse)

We can describe waves mathematically using the following formulas:

Wave moves toward +x direction: [pic]

Wave moves toward -x direction: [pic]

- 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: [pic] (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: [pic] (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: [pic] (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): [pic]

- 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: [pic] Red-shifted

Light from approaching source: [pic] 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. [pic] (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):

[pic] [pic] [pic] (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):

[pic] [pic] (m = 1, 2, 3, …)

Longitudinal standing waves (tube open one end and closed at other end):

[pic] [pic] (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 m = 0, 1, 2, …

For identical sources, ΔΦ0 =0 rad, so Δx = mλ

Perfect destructive interference: ΔΦ = 2πΔx/λ + ΔΦ0 = (m+ ½) 2π rad m = 0, 1, 2, …

For identical sources, ΔΦ0 =0 rad, so Δx = (m + ½) λ

Thin Film Interference (21.6)

Index of Refraction: [pic] (n ≥ 1)

Wavelength of light in a material: [pic]

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: [pic]

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 m = 0, 1, 2, …

For identical sources, ΔΦ0 =0 rad, so Δr = mλ

Perfect destructive interference: ΔΦ = 2πΔr/λ + ΔΦ0 = (m+ ½) 2π rad m = 0, 1, 2, …

For identical sources, ΔΦ0 =0 rad, so Δr = (m + ½) λ

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: [pic]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

[pic]

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: [pic]

α = coefficient of linear expansion; units = 1/Cº = (Cº)-1)

Use this formula for one-dimensional expansion (or contraction) problems. The one-dimension could be length, or it could be radius.

Volume Thermal Expansion: [pic]

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: [pic]

n = number of moles of gas

R = universal gas constant = 8.31 J/(mol K)

T = temperature in Kelvins

Alternative version: [pic]

N = number of gas particles

k = Boltzmann’s constant = 1.38x10-23 J/(K molecules)

Boyle’s Law (constant T, constant n): [pic]

Charles’ Law (constant P, constant n): [pic]

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

[pic] (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

[pic]

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: [pic]

c = specific heat capacity; units = J/(kg K)

Heat capacity of a material in terms of molar specific heat capacity, C: Q = CnΔT

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: [pic]

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

[pic] where [pic]

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: [pic]

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: [pic]

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:

[pic]

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

[pic]

Pressure in a gas (18.2)

Root mean square speed, vrms = sqrt((vx2)avg + (vy2)avg + (vz2)avg)

[pic]

Temperature (18.3)

Average translational kinetic energy of a molecule: [pic]

[pic]

Temperature measures the average translational kinetic energy of a gas

[pic] T in Kelvins!

[pic] 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: [pic]

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: [pic][pic]= 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.

[pic][pic] TC, and TH must be in Kelvins.

Efficiency of Carnot engine: [pic][pic]

Carnot coefficient of performance: KCarnot = TC/(TH-TC)

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download