PH605

Thermal and Statistical Physics

M.J.D.Mallett P.Blümler

Recommended text books: • • •

Finn C.B.P. : Adkins C.J. : Mandl F:

Thermal Physics Equilibrium Thermodynamics Statistical Physics

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THERMODYNAMICS .....................................................................................................4 Review of Zeroth, First, Second and Third Laws...................................................4 Thermodynamics................................................................................................4 The zeroth law of thermodynamics,...................................................................4 Temperature, T...................................................................................................4 Heat, Q ...............................................................................................................4 Work, W.............................................................................................................4 Internal energy, U ..............................................................................................5 The first law of thermodynamics, ......................................................................5 Isothermal and Adiabatic Expansion .................................................................6 Heat Capacity.....................................................................................................6 Heat capacity at constant volume, CV ................................................................7 Heat capacity at constant pressure, CP ...............................................................7 Relationship between CV and CP .......................................................................8 The second law of thermodynamics, .................................................................8 Heat Engines ......................................................................................................9 Efficiency of a heat engine ..............................................................................10 The Carnot Cycle .............................................................................................11 The Otto Cycle .................................................................................................13 Concept of Entropy : relation to disorder............................................................15 The definition of Entropy.................................................................................16 Entropy related to heat capacity.......................................................................16 The entropy of a rubber band...........................................................................17 The third law of thermodynamics, ...................................................................18 The central equation of thermodynamics.........................................................18 The entropy of an ideal gas..............................................................................18 Thermodynamic Potentials : internal energy, enthalpy, Helmholtz and Gibbs functions, chemical potential ...............................................................................19 Internal energy .................................................................................................20 Enthalpy ...........................................................................................................20 Helmholtz free energy......................................................................................20 Gibbs free energy .............................................................................................21 Useful work......................................................................................................21 Chemical Potential ...........................................................................................22 The state functions in terms of each other .......................................................22 Differential relationships : the Maxwell relations...............................................23 Maxwell relation from U .................................................................................23 Maxwell relation from H .................................................................................24 Maxwell relation from F ..................................................................................24 Maxwell relation from G .................................................................................25 Use of the Maxwell Relations..........................................................................26 Applications to simple systems.............................................................................26 The thermodynamic derivation of Stefan’s Law .............................................27 Equilibrium conditions : phase changes..............................................................28 Phase changes ..................................................................................................28 P-T Diagrams ...................................................................................................29 PVT Surface.....................................................................................................29 First-Order phase change .................................................................................30 Second-Order phase change.............................................................................31

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Phase change caused by ice skates...................................................................31 The Clausius-Clayperon Equation for 1st order phase changes. ......................32 The Ehrenfest equation for 2nd order phase changes .......................................33 BASIC STATISTICAL CONCEPTS ..................................................................................35 Isolated systems and the microcanonical ensemble : the Boltzmann-Planck Entropy formula ...................................................................................................35 Why do we need statistical physics ?...............................................................35 Macrostates and Microstates............................................................................35 Classical vs Quantum.......................................................................................36 The thermodynamic probability, Ω..................................................................36 How many microstates ?..................................................................................36 What is an ensemble ?......................................................................................37 Stirling’s Approximation .................................................................................39 Entropy and probability.......................................................................................39 The Boltzmann-Planck entropy formula..........................................................40 Entropy related to probability ..........................................................................40 The Schottky defect .........................................................................................41 Spin half systems and paramagnetism in solids...................................................43 Systems in thermal equilibrium and the canonical ensemble : the Boltzmann distribution...........................................................................................................45 The Boltzmann distribution .............................................................................45 Single particle partition function, Z, and ZN for localised particles : relation to Helmholtz function and other thermodynamic parameters .................................47 The single particle partition function, Z ..........................................................47 The partition function for localised particles ...................................................47 The N-particle partition function for distinguishable particles........................47 The N-particle partition function for indistinguishable particles.....................48 Helmholtz function ..........................................................................................49 Adiabatic cooling .............................................................................................50 Thermodynamic parameters in terms of Z.......................................................53

PH605 : Thermal and Statistical Physics

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Thermodynamics Review of Zeroth, First, Second and Third Laws Thermodynamics Why study thermal and statistical physics ? What use is it ?

The zeroth law of thermodynamics, If each of two systems is in thermal equilibrium with a third, then they are also in thermal equilibrium with each other. This implies the existence of a property called temperature. Two systems that are in thermal equilibrium with each other must have the same temperature. Temperature, T th

The 0 law of thermodynamics implies the existence of a property of a system which we shall call temperature, T. Heat, Q In general terms this is an amount of energy that is supplied to or removed from a system. When a system absorbs or rejects heat the state of the system must change to accommodate it. This will lead to a change in one or more of the thermodynamic parameters of the system e.g. the temperature, T, the volume, V, the pressure, P, etc. Work, W When a system has work done on it, or if it does work itself, then there is a flow of energy either into or out of the system. This will also lead to a change in one or more of the thermodynamics parameters of the system in the same way that gaining or losing heat, Q, will cause a change in the state of the system, so too will a change in the work, W, done on or by the system. When dealing with gases, the work done is usually related to a change in the volume, dV, of the gas. This is particularly apparent in a machine such as a cars engine.

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If a thermally isolated system is brought from one equilibrium state to another, the work necessary to achieve this change is independent of the process used. We can write this as,

dU = đ WAdiabatic

Note : when we consider work done we have to decide on a sign convention. By convention, work done on a system (energy gain by the system) is positive and work done by the system (loss of energy by the system) is negative. e.g. Internal energy, U The internal energy of a system is a measure of the total energy of the system. If it were possible we could measure the position and velocity of every particle of the system and calculate the total energy by summing up the individual kinetic and potential energies. N

N

n =1

n =1

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đ W = + PdV : compression of gas in a pump (T of gas increases). đ W = − PdV : expansion of gas in an engine (T of gas decreases).

Isothermal and Adiabatic Expansion When we consider a gas expanding, there are two ways in which this can occur, isothermally or adiabatically.

U = ∑ KE + ∑ PE However, this is not possible, so we are never able to measure the internal energy of a system. What we can do is to measure a change in the internal energy by recording the amount of energy either entering or leaving a system.

•

Isothermal expansion : as it’s name implies this is when a gas expands or contracts at a constant temperature (‘iso’-same, ‘therm’temperature). This can only occur if heat is absorbed or rejected by the gas, respectively. The final and initial states of the system will be at the same temperature.

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Adiabatic expansion : this is what happens when no heat is allowed to enter or leave the system as it expands or contracts. The final and initial states of the system will be at different temperatures.

In general, when studying thermodynamics, we are interested in changes of state of a system.

∆U = ∆Q + ∆W which we usually write,

dU = đ Q + đ W

The bar through the differential, đ , means that the differential is inexact, this means that the differential is path dependent i.e. the actual value depends on the route taken, not just the start and finish points.

Heat Capacity As a system absorbs heat it changes its state (e.g. P,V,T) but different systems behave individually as they absorb the same heat so there must be a parameter governing the heat absorption, this is known as the heat capacity, C. The first law of thermodynamics,

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The heat capacity of a material is defined as the limiting ration of the heat, Q, absorbed, to the rise in temperature, ∆T, of the material. It is a measure of the amount of heat required to increase the temperature of a system by a given amount.

PH605 : Thermal and Statistical Physics Relationship between CV and CP The internal energy of a system can be written as,

dU = đ Q + đ W

Q C = limit ∆T → 0 ∆T

⇒ đ Q = dU - PdV

When a system absorbs heat its state changes to accommodate the increase of internal energy, therefore we have to consider how the heat capacity of a system is governed when there are restrictions placed upon how the system can change.

Assuming the change of internal energy is a function of volume and temperature, U = U (V , T ) , i.e. we have a constant pressure process, this can be written as,

∂U ∂U đQ = dV + dT + PdV ∂V T ∂T V

In general we consider systems kept at constant volume and constant temperature and investigate the heat capacities for these two cases. which leads to,

Heat capacity at constant volume, CV If the volume of the system is kept fixed then no work is done and the heat capacity can be written as,

CV =

đQV ∂U = dT ∂T V

đ QP ∂U ∂V ∂U ∂V = + + P dT ∂V T ∂T P ∂T V ∂T P ∂U ∂V ∴CP = CV + P + ∂V T ∂T P

⇒ CP =

This is the general relationship between CV and CP. In the case of an ideal gas the internal energy is independent of the volume (there is zero interaction between gas particles), so the formula simplifies to,

Heat capacity at constant pressure, CP The heat capacity at constant pressure is therefore analogously,

CP =

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∂V CP = CV + P ∂T P ⇒ CP − CV = R

đQP dT

We now use a new state function known as enthalpy, H, (which we discuss later).

H = U + PV ⇒ dH = dU + PdV + VdP dH = đ Q + VdP

The second law of thermodynamics, The Kelvin statement of the 2nd law can be written as, It is impossible to construct a device that, operating in a cycle, will produce no effect other than the extraction of heat from a single body at a uniform temperature and the performance of an equivalent amount of work.

Using this definition we can write,

CP =

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đQP ∂H = dT ∂T P

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It would be useful to convert all the heat , QH, extracted into useful work but this is disallowed by the 2nd law of thermodynamics.

A more concise form of this statement is, A process whose only effect is the complete conversion of heat into work is impossible. Another form of the 2nd law is known as the Clausius statement,

If this process were possible it would be possible to join two heat engines together, whose sole effect was the transport of heat from a cold reservoir to a hot reservoir.

It is impossible to construct a device that, operating in a cycle, will produce no effect other than the transfer of heat from a colder to a hotter body.

Efficiency of a heat engine Heat Engines Heat engines convert internal energy to mechanical energy. We can consider taking heat QH from a hot reservoir at temperature TH and using it to do useful work W, whilst discarding heat QC to a cold reservoir TC.

We can define the efficiency of a heat engine as the ratio of the work done to the heat extracted from the hot reservoir.

η=

W QH − QC Q = = 1− C QH QH QH

From the definition of the absolute temperature scale1, we have the relationship,

QC QH = TC TH

1 For a proof of this see Finn CP, Thermal Physics, [email protected]

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One way of demonstrating this result is the following. Consider two heat engines which share a common heat reservoir. Engine 1 operates between T1 and T2 and engine 2 operates between T2 and T3. We can say that there must be a relationship between the ratio of the heat extracted/absorbed to the temperature difference between the two reservoirs, i.e.

Q = f (θ ,θ Q 1

1

2

)

Q = f (θ ,θ Q

,

'

2

2

2

3

)

Q = f (θ ,θ Q

,

1

''

1

3

3

)

3

Therefore the overall heat engine can be considered as a combination of the two individual engines.

f (θ ,θ ''

1

) = f (θ ,θ ) f (θ '

3

1

2

2

,θ

3

)

However this can only be true if the functions factorize as,

f (θ x ,θ y ) →

The Carnot cycle is a closed cycle which extracts heat QH from a hot reservoir and discards heat QC into a cold reservoir while doing useful work, W. The cycle operates around the cycle A►B►C►D►A

T (θ x )

T (θ y )

Where T(θ) represents a function of absolute, or thermodynamic temperature. Therefore we have the relationship,

Q1 T (θ1 ) = Q2 T (θ 2 ) Therefore we can also write the efficiency relation as,

η = 1−

TC TH

The efficiency of a reversible heat engine depends upon the temperatures between which it operates. The efficiency is always