Physics 112 Ucsc

Front 1

Ideal Gas Law

Back 1

PV= nRT=NkT

Front 2

v_rms (root-mean-squared velocity)

Back 2

sqrt[3kT/m]

Front 3

U_thermal

Back 3

N*f*kT/2 where f is degrees of freedom

Front 4

First Law of Themodynamics

Back 4

▲U = Q + W

Front 5

"quasistatic"

Back 5

a volume change process slow enough that the gas is continually in equilibrium during the process

Front 6

Quasistatic Work

Back 6

W= – P▲V for constant pressure, or
W= – ∫P(V) dV from V_i to V_f if P(V) known

Front 7

isothermal

Back 7

▲T=0 —>▲U = 0 . Often this process is slow enough to treat quasistatically for calculating work.

Front 8

Adiabatic

Back 8

Q=0, no Heat —>▲U = W

PV^(gamma) = constant ( use to solve work by treating as quasistatic)

Front 9

Adiabatic exponent (\gamma)

Back 9

\gamma = (f+2)/f where f is the degrees of freedom

Front 10

heat capacity

Back 10

heat per degree temperature increase,
C ≡ Q/▲T =(▲U-W)/▲T

Front 11

Heat capacity at constant volume

Back 11

constant volume implies zero work thus,

C_V = (▲U/▲T)_v = (∂U/ ∂T)_v for infinitesimal

change where _v indicates constant volume

Front 12

Heat capacity at constant pressure

Back 12

C_p = ([▲U - (-P▲V)] /▲T)_p
= (∂U/∂T)_p+P(∂V/∂T)_p

For ideal gas, (∂U/∂T)_p= (∂U/∂T)_v thus
C_p = C_v+Nk

Front 13

Microstate

Back 13

One possible state

Front 14

Macrostate

Back 14

the number of unique states, not counting degeneracies

Front 15

Multiplicity

Back 15

the number of degenerate Microstates corresponding to a given Macrostate

Front 16

Paramagnet

Back 16

a material which is magnetically active and aligns parallel to any external B-field

Front 17

Einstein Solid

Back 17

a model of solids in which each atom is 3 seperate yet identical oscillators with quantized energy

Front 18

Fundamental assumption of Stat Mech

Back 18

In an isolated system in thermal equilibrium, all accessible microstates are equally probable.

Front 19

Stirling's Approximation (including the pi, his only contribution)

Back 19

ln(N!)= NlnN - N + ½ln(πN) or

N!= N^N*e^(–N)*sqrt (πN)

Front 20

Second Law of Thermodynamics

Back 20

"Multiplicity tends to increase."
"Entropy tends to increase."
"You can't win."

Front 21

Irreversible and Reversible

Back 21

Irreversible processes create new Entropy. Isentropic (▲S = 0) implies a Reversible process because no entropy was created, so reversing the process doesn't violate 2nd Law

Front 22

Entropy

Back 22

S ≡ k ln Ω

S= -(∂F/∂T)_V,N

S= -(∂G/∂T)_P,N

Front 23

Temperature

Back 23

1/T ≡ (∂S/∂U)_N,V

Temperature is the quantity that's the same in thermal equilibrium.

Front 24

Pressure

Back 24

P ≡= T (∂S/∂V)_U,N

Pressure is the quantity that is the same for systems in mechanical equilibrium.

Front 25

Chemical Potential ( µ )

Back 25

µ ≡ – T (∂S/∂N)_U,V


Quantity that's the same for systems in diffusive equilibrium (particles being shared)

Front 26

change in Entropy

Back 26

dS = dU/T = Q/T for a quasistatic process (or when W=0)

Often written ▲S = Q/T when finite change is considered under constant temperature. Otherwise
dS = C_v*dT/T if temperature is varying, and you integrate to get ▲S

Front 27

Third Law of Thermodynamics

Back 27

C_v –> 0 as T –> 0
As T –> 0, S –> S(0), residual entropy

A more conceptual formulation is given as follows: The rate change in entropy goes to 0 as T goes to 0, thus S=0 is unattainable in finite time intervals (read reality), thus T=0 K is impossible.

Front 28

Magnetization

Back 28

M = - U/B where B is the magnitude of the B-field

Front 29

Quasistatic Heat

Back 29

Q = Tds for a quasistatic process

Front 30

Thermodynamic Identity (IMPORTANT)

Back 30

dU = TdS - PdV + µdN

Setting various things to 0 (dt= 0 etc) also you to rederive the formulas for these three important quantities, T, P, and µ.

Front 31

Efficiency of an Engine ( e )

Back 31

e = W/ Q_h where Q_h is the heat absorbed.

More useful though, using W = Q_h - Q_c,

e = 1 - Q_c/Q_h where Q_c is the heat expelled to the cold reservoir. Entropy considerations give the inequality

e ≤ 1 – T_c/T_h

Front 32

Work in an Engine

Back 32

Q_h=W+Q_c with both Q's taken to be positive

Front 33

Carnot Cycle

Back 33

A cycle of two adiabats and two isotherms that achieves maximum theoretical value of efficiency, e = 1 – T_c/T_h

Front 34

Coefficient of Performance (COP)

Back 34

COP = Q_c/W = 1/(Q_h/Q_c – 1) . Used to rate Refrigeration processes.

Front 35

Curie's Law

Back 35

M ∝ 1/T which holds at high T limits for all paramagnets.

Front 36

Rankine Cycle

Back 36

Standard cycle utilized in Steam Engines

Front 37

Otto Cycle

Back 37

4 and 2 Strokes Engines use this cycle where a gas is adiabatically compressed, ignited by a spark(plug) pushing the piston during expansion, waste is expelled and new fuel injected. Rinse repeat

Front 38

Diesel Cycle

Back 38

Compress air to temperatures hot enough such that upon injection of fuel, it immediately ignites pushing the piston. Generally tuned to maintain near constant pressure throughout the cycle.

Front 39

Throttling

Back 39

A process where a gas is forced through a porous plug or a very small valve . Enthalpy remains constant during this process.

The attractive potential (-U_potential) generally dominates the repulsive (see exceptions). It's negative , but becomes less negative as pressure drops and the distance between molecules increases. Energy conservation requires the kinetic energy thus drop to compensate, which cools the gas strongly.

Front 40

Exceptions to Throttling

Back 40

Throttling processes (Hampson-Linde Cycle) liquefies most gases except Hydrogen and Helium, which will only liquefy if drastically cooled before the throttling process. Intermolecular attraction for these two gases is much too small, collisions dominate INCREASING temperature in this process unless molecules are moving sufficiently slow (read cold).

Front 41

Laser Cooling

Back 41

Absolutely badass.

Basically a low density cloud of atoms is shot at by a laser of frequency slightly below the threshold within which the atoms will absorb the photons. Atoms moving toward the laser see doppler shifted light and absorb a photon, then spontaneously emit it. In this process momentum is gained. Shine Laser from every direction, and you oppose motion in every direction, causing cooling.

Front 42

Enthalpy, H

Back 42

H= U+ PV

Total Energy needed to create a system, and displace the environment to create the space it occupies.

Front 43

Helmholtz free Energy

Back 43

F= U- TS

F= -kTlnZ as well

Total work needed to create a system out of nothing (environment provides some energy for free)

Front 44

Gibbs Free Energy

Back 44

G= U-TS+PV

Total work you need to do to create a system out of nothing

Front 45

Thermodynamic Potentials (little memory trick)

Back 45

Remember, you can derive all the relevant partial derivative relations from the Thermo Identities by setting stuff to 0!

Front 46

"Free Energy is a Force toward Equilibrium"

Back 46

If no particles can enter or leave a system

*At constant energy and volume, S tends to increase (maximized by system)

*At constant temperature and volume, F tends to decrease (minimized by system)
*At constant temperature and pressure, G tends to decrease (minimized by system)

Front 47

Extensive Quantity

Back 47

V, N, S, U, H, F, G, mass

If you double the amount of "stuff" these double. If you multiple Extensive quantity with intensive, you get an extensive quantity. Ex/Ex = Intensive Quantity. Ex*Ex = nonsense your doing it wrong. Adding changes nothing but adding Ex+In is again nonsense

Front 48

Intensive

Back 48

T, P, µ, density

These things don't double when the amount of "stuff" is doubled.

Front 49

Interpretation of µ (Chemical Potential)

Back 49

G = Nµ , chemical potential is just the Gibbs free energy per particle!

Front 50

Van Der Waals Equation of state
(description not formula)

Back 50

a qualitative model for understanding real gases. The first correction, a, accounts for short range attractive forces ("stickiness") while the second correction accounts for for the strong short range repulsive forces (molecules have "finite volume" and gas isn't infinitely compressible)

Front 51

Boltzmann factor

Back 51

exp[-ßE] (that's a shitty Beta)

ß= 1/kT

Front 52

Probability in Boltzmann Stats

Back 52

P(s) = 1/Z exp[-ßE(s)]

where Z is the partition function, the sum of all Boltzmann factors

Front 53

Average Energy( E-bar)

Back 53

-1/Z ∂Z/∂ß = -∂/∂ßlnZ

U= NE-bar, so depending on Z, this formula can also yield total energy U (this is up to whether you put the N in the power of Z, or multiply it in later)

Front 54

Standard Deviation

Back 54

where E^2-bar is given by taking another derivative with respect to beta (given by Johnson)

Front 55

Partition Function for noninteracting, indistinguishable particles

Back 55

Resolution to Gibb's Paradox, truly indistinguishable particles

Front 56

Gibbs Factor

Back 56

Like Boltzmann, but allows for particle exchange with reservoir.

Front 57

Bosons

Back 57

Photons, pions, etc. Integer-spin particles that can occupy the same state. Generally all the boson try to get into the lowest available energy state

Front 58

Fermions

Back 58

Electrons, Protons, Neutrons, Neutrinos etc. Half-integer spin particles. These can never occupy the same state due to Pauli Exclusion (well that's what its called at least, which is hardly telling of the physics involved).

Front 59

Fermi-Dirac Distribution

Back 59

Front 60

Bose-Einstein Distribution

Back 60

Front 61

Fermi Energy

Back 61

At T=0, n_FD becomes a step function with the cutoff at µ(T=0), which we call the fermi energy. A gas like this is said to be Degenerate

Front 62

Bulk Modulus

Back 62

A useful quantity that's easily measured, goes at the change in pressure when the material in compressed, divided by the change in volume:

Front 63

Density of States

Back 63

dn/dε : find surface area of 1/8 sphere in momentum space, multiply by dp, divide by the volume of a single momentum state (pi*h/2L)^3. This essentially tells you how many states there are if you go from p to p+dp

Front 64

Ultraviolet Catastrophe

Back 64

Basically before well developed QM, because the number of possible wavelengths in an electromagnetic wave is infinite, shouldn't the total thermal energy produced by all these waves be infinite? Of course not, QM tells us that energy of a harmonic oscillator is quantized, and thus higher energy (shorter wavelengths) get exponentially suppressed.

Front 65

Planck distribution

Back 65

Essentially the Bose-Einstein Distribution with µ=0 and ε=hf. This makes sense because photons are Bosons, and they can be created and destroyed, thus µ must be 0

Front 66

Photons

Back 66

Spin 1 Bosons. The particles of light. Essentially each photon is a quantum harmonic oscillator that moves through space. Photons have 2 polarizations, so make sure to slip in that factor of 2 when you find U or N from the distribution

Front 67

Wien's Law

Back 67

Higher temperatures yield higher energy photons.

Front 68

Stefans' Law

Back 68

P/A = σT^4

where σ is Stefan-Boltzmann constant "5-6-7-8" 5.67x 10^-8

Front 69

Blackbody Radiation

Back 69

Radiation emmited by a non-reflective object. Completely equivalent to the emission of radiation from a hole in a box of radiation. (Essentially Stefan's Law)

Front 70

Emissivity (e)

Back 70

A measure of how reflective a surface is. 1 is a perfect blackbody, 0 is a perfectly reflective surface. Modifies Stefan's Law:

P/A =σeT^4

Front 71

Phonons

Back 71

Spin 0 Bosons (unless Transverse). "Particles" of oscillation in a solid. Essentially these are embodiments of the modes of oscillation of coupled oscillators. Waves in solid analogous to waves of light, phonons analogous to photons.

Front 72

Debye Approximation

Back 72

Peter Debye basically said that summing over arbitrary shapes of the crystal lattice (which determines the shape of "n-space") is too hard, so let's pretend its a sphere, convert the sum to an integral in spherical cords and bam, its solvable. Magically, this approximation is exact in both low and high temperature limits.

Front 73

Bose-Einstein Condensation (BEC)

Back 73

The tendency for bosons below a certain temperature (called the condensation temperature T_c) to aburtly accumulate in the ground-state (lowest energy state). At low temperatures, µ is very small, and ε_0 is very small, so the distribution favors ground state (ridiculously favors).

Front 74

Superfluid

Back 74

The superfluid component is an example of a BEC. The superfluid atoms are constantly moving to the lowest available state, and so behave in mind-blowing ways.

Front 75

Curie Temperature

Back 75

Temperature at which a ferromagnet's net magnetization becomes zero.

Front 76

Ising Model

Back 76

Model of a ferromagnet in which long-range magnetic interaction between dipoles is ignored, and the assumptions that there is a preferred axis of magnetization and that each dipole can only point parallel or antiparallel to it are made.

Front 77

Mean Field Approximation

Back 77

The Approximation here is that at every moment, the alignments of all dipoles is "typical" - no major fluctuations that cause M to be less or more than average

Front 78

Monte-Carlo Simulation

Back 78

A computer simulations where a Metropolis algorithm(give each dipole a ton of opportunities to flip) is used in combination with importance sampling (probability to flip weighted on exp(-ΔU/kT) to try and model the behavior of a ferromagnet. Quite successful if done in enough complication.