Fundamentals and Entropy Flashcards

1
Q

Fundamentals

Classical Thermodynamics - First Law

A

dE = TdS - pdV

-this implies that any process in which energy is conserved is possible

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2
Q

Fundamentals

Classical Thermodynamics - Second Law

A

dS = dQrev / T
rev = reversible
-the entropy of an isolated system can only remain constant or increase
-this implies that heat cannot flow from a cold to a hot reservoir without other changes

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3
Q

Fundamentals

Direction of Time

A
  • hot and cold cannot spontaneously separate

- entropy increases with time for an isolated system, or for the universe as a whole

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4
Q

Fundamentals

Statistical Physics

A
  • the macroscopic properties of a system are deduced from the statistical behaviour of the constituent particles
  • to do this we need to consider large systems, but 1mm³ of air contains ~10^16 molecules
  • all states with equal energy are equally probable
  • increasing the energy of a state decreases its probability
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5
Q

Microstate

Definition

A

-specified by describing the (quantum) state of each particle in the system

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6
Q

Macrostate

Definition

A
  • properties of a system as a whole i.e. T, p, V, E< G etc.
  • no information on individual particles
  • every macrostate has an enumerable number of microstates
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7
Q

Postulate of equal a priori probabilities

A

-for a given macrostate, all microstates are equally probable

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8
Q

The Ergodic Hypothesis

A
  • the time average of a system is equal to the instantaneous ensemble average
  • i.e. the average of many identical measurements made on a single system is equal to the average of a single measurement on many copies of the system
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9
Q

Multiplicity

Definition

A
  • the number of microstates in a macrostate

- for each macrostate there is a large number of macroscopically indistinguishable microstates

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10
Q

What does the probability of a macrostate depend on?

A
  • the probability Pi of a certain macrostate depends on how many microstates correspond to the macrostate
  • the probability of each microstate is proportional to its multiplicity
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11
Q

Probability of a Macrostate

Equation

A

Pi = ωi / Σωi

-where the sum is over all microstates in the system

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12
Q

Addition Rule

A

-for mutually exclusive events

P(i or j) = P(i) + P(j)

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13
Q

Multiplication Rule

A

-for independent events

P(i and j) = P(i) x P(j)

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14
Q

Distinguishing Between Different Macrostates

A

-different macrostates can be distinguished by e.g. weight

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15
Q

Distinguishing Between Different Microstates

A

-different microstates that correspond to the same macrostate cannot be distinguished between at the macroscopic level e.g. by weight

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16
Q

Boltzmann’s Expression for Entropy

Equation

A

S = k ln Ω

S = entropy
Ω = multiplicity
17
Q

Boltzmann’s Expression for Entropy

Calculating Entropy

A
  • you need to know Ω to calculate entropy
  • probability is very sharply peaked around the most probable state
  • the greater the number of states, N, the sharper the peak
  • the distribution tends to a Gaussian as N->∞
18
Q

Boltzmann’s Expression for Entropy

Fluctuations

A

fractional magnitude of fluctuations ∝ 1/√N

  • as N is increased, the fluctuations become proportionally smaller and smaller
  • for macroscopic systems (in equilibrium) fluctuations are so small they are unimportant
19
Q

Stirling’s Approximation

A

-for a macroscopic system of N≈10^20, the multiplicity N! is massive
-to deal with numbers like this, take their natural logarithm:
ln(N!) = ln1 + ln2 + .. + lnN
≈ 1,N ∫ln(x) dx
= [ xlnx - x ] N,1
≈ N lnN - N

20
Q

Total Multiplicity vs Maximum Multiplicity

A
  • we can replace total multiplicity Ω=2^N by maximum multiplicity Pmax = N! / (N/2)!(N/2)!
  • this makes a negligible difference, especially with the natural logs
21
Q

Origin of Irreversibility

Classical Thermodynamics

A
  • consider a container with a gas partitioned in one half, when the partition is removed the gas will move to fill the container
  • so V1 -> V2 where V2=2*V1
  • > ΔS = nR ln(V2/V1) = Nkln2
22
Q

Origin of Irreversibility

Statistically

A

-consider a container with a gas partitioned in one half, when the partition is removed the gas will move to fill the container
-there are N particles contained in half of the containers volume
-when the partition is removed, the number of states available to each particle effectively doubles
-so the multiplicity increases 2^N times
-> ΔS = Nk ln(2Ω) - Nk lnΩ
= Nkln2

23
Q

Origin of Irreversibility

Probability that all atoms end up on one side

A
P = (1/2)^N = 1/(2^N)
-let N = 3.3x10^20, the number of molecules in a cubic centimetre of air
log(2^N) = N log22 = 0.30N
P = 1/10^(0.3N) = 10^(-10^20)
-the probability that all particles are on the left side of the container is effectively zero
24
Q

Nernst’s Therorem

A

-any entropy changes in an isothermal reversible process approach zero as the temperature approaches zero
T->0 lim ΔSrev = 0

25
Third Law
- it is impossible by any procedure to reduce any system to the zero of temperature of a finite number of steps - it is impossible by any procedure to reduce the entropy of a system to its zero-point value in a finite number of steps
26
Typical Way to Reduce Temperature
-to alternate between isothermal and adiabatic expansions
27
Entropy at Absolute Zero
- a system at T=0 is in its ground state so that its entropy is determined only by the degeneracy pf the ground state - only systems such as elementary (perfect) crystals have S(0) = 0 (Ω=0) - S(0) is not usually zero, there is usually some residual entropy
28
Residual Entropy
- may come from a multiplicity of possible molecular orientations in the crystal (degeneracy of the ground state) even at absolute zero - the third law applies to equilibrium states only, glasses are not in equilibrium as the relaxation time is huge, glasses have particularly large entropy at T=0
29
Clausius-Clapeyron Equation
∂p/∂T = ΔS/ΔV
31
Consequences of the Third Law
-clausius-clapeyron equation: ∂p/∂T = ΔS/ΔV -so ΔS->0 implies: ∂p/∂T->0 as T->0 -on a P-T diagram the line should asymptotically approach the x-axis at T=0
32
Heat Capacity at Absolute Zero
0,T ∫ 𝛿Q/T' = St - So = 0,T ∫ Cp(T')/T' dT' => Cp(0) = 0 -to avoid divergence at T=0
32
Volume Coefficient of Thermal Expansion at Absolute Zero
``` α = 1/V [∂V/∂T] |p -using Maxwell's Relation α = - 1/V [∂S/∂p] |p -since [∂S/∂p] |p -> 0, α->0 as T->0 ```
33
What is the equation that will give you the number of ways of picking n objects from N objects?
= N! / n!(N-n)!