Fundamentals and Entropy Flashcards
Fundamentals
Classical Thermodynamics - First Law
dE = TdS - pdV
-this implies that any process in which energy is conserved is possible
Fundamentals
Classical Thermodynamics - Second Law
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
Fundamentals
Direction of Time
- hot and cold cannot spontaneously separate
- entropy increases with time for an isolated system, or for the universe as a whole
Fundamentals
Statistical Physics
- 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
Microstate
Definition
-specified by describing the (quantum) state of each particle in the system
Macrostate
Definition
- 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
Postulate of equal a priori probabilities
-for a given macrostate, all microstates are equally probable
The Ergodic Hypothesis
- 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
Multiplicity
Definition
- the number of microstates in a macrostate
- for each macrostate there is a large number of macroscopically indistinguishable microstates
What does the probability of a macrostate depend on?
- 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
Probability of a Macrostate
Equation
Pi = ωi / Σωi
-where the sum is over all microstates in the system
Addition Rule
-for mutually exclusive events
P(i or j) = P(i) + P(j)
Multiplication Rule
-for independent events
P(i and j) = P(i) x P(j)
Distinguishing Between Different Macrostates
-different macrostates can be distinguished by e.g. weight
Distinguishing Between Different Microstates
-different microstates that correspond to the same macrostate cannot be distinguished between at the macroscopic level e.g. by weight
Boltzmann’s Expression for Entropy
Equation
S = k ln Ω
S = entropy Ω = multiplicity
Boltzmann’s Expression for Entropy
Calculating Entropy
- 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->∞
Boltzmann’s Expression for Entropy
Fluctuations
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
Stirling’s Approximation
-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
Total Multiplicity vs Maximum Multiplicity
- 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
Origin of Irreversibility
Classical Thermodynamics
- 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
Origin of Irreversibility
Statistically
-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
Origin of Irreversibility
Probability that all atoms end up on one side
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
Nernst’s Therorem
-any entropy changes in an isothermal reversible process approach zero as the temperature approaches zero
T->0 lim ΔSrev = 0
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
Typical Way to Reduce Temperature
-to alternate between isothermal and adiabatic expansions
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
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
Clausius-Clapeyron Equation
∂p/∂T = ΔS/ΔV
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
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
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
What is the equation that will give you the number of ways of picking n objects from N objects?
= N! / n!(N-n)!
