Thermal Physics Introduction

Rengachary Parthasarathy Jan 23,2013

How Large is Large?

• What is the temperature in this room? Say 200C. • The temperature is due to the air molecules. How many air molecules are in this room? • Estimate the size of this room, say 10m x 10m x 3m; vol = 300m3 • 1L = 1000cm3 ; 1m3 = 100cm x100cm x 100cm • = 1000000cm3 = 103L • Vol = 300 x103L = 3 x 105L

How Large is Large?

• At STP, I mole of air (ideal gas) occupies 22.4 L. • At 200C , the number of moles of air particles (using PV = nRT) is • n = (1atm x 3x105L)/(0.0820L.atm/mol- 1.K)(293.15)K • = 3x105 atm.L /(24.03 atm.L.mole-1) • = 0.125 x 105 moles • = 6.02 x 1023 x 0.125 x 105 molecules! • = 7.51 x 1027 molecules! • Not a chance! But we assume they move randomly and apply the laws of probability to predict how they behave as a whole.

• Some of the properties of bulk matter does not depend on the details. • Example: Heat flows from hot to cold spontaneously • The properties of bulk matter that does not depend on the microscopic details and the principles that generalize them comprises thermodynamics. • Examples: Liquids boil more readily at lower pressure. • The maximum efficiency of heat engine is the same whether steam or air is used. • How to make connection between one molecule and 10 23 ?

• Do we want to understand matter in more detail?

• For this, we must take into account the quantum behavior of atoms and the laws of statistics. • When we do, we can explain the properties of materials, say metals. • We can also explain why the principles of thermodynamics are what they are-why heat flows from hot to cold, for example. • This underlying explanation of thermodynamics comes from statistical mechanics.

• “Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two little points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it does not bother you anymore” • Arnold Sommerfeld • “Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two little points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it does not bother you anymore” • Arnold Sommerfeld • Classical Thermodynamics is the only physical theory of universal content which will never be overthrown • Einstein Thermal Physics = Thermodynamics + Statistical Mechanics • Conceptually, a difficult part of the under- graduate physics program. • Thermodynamics is concerned with macroscopic properties of a system • Statistical mechanics is the bridge between the microscopic and macroscopic worlds

Chapter 1. Energy in Thermal Physics • Thermal equilibrium • What is the most familiar concept in thermodynamics?

• Temperature! • But wait, what is temperature?

• Simple! Temperature is what you measure with a thermometer! • The above definition is an opera- tional definition. Tells you how to measure temperature. • Why this procedure works? • Because the thermometer relies on the fundamental fact: When two objects are in contact with each other, they eventually tend to come to the same temperature. • This fact is so fundamental that we can use this as a theoretical definition of temperature. • Temperature is the thing that is the same for two objects after they have been in contact long enough. • Is this not vague? What kind of contact? How long? • Is there more than one quantity that ends up being the same for both objects? • After two objects have been in contact long enough, they are in thermal equilibrium. • Relxation time is the time required for a system to come to thermal equilibrium. • Even if we try to prevent the two objects to come to equilibrium, we can not; eventually, they do. All we have done is to increase the relxation time.

• Contact: Some means for the two objects to exchange energy spontaneously, in the form of heat. • Contact: some means for the two objects to exchange energy spontaneously in the form of “heat”. • Conduction, convection and radiation

• We have to refine our ideas of equilibrium. Pour cold cream in a cup of hot coffee. • In a few seconds of relaxation time, the coffee and the cream come to the same temperature- reach equilibrium. • But it takes much longer for the coffee to come to equilibrium with its surroundings. • The cream and the coffee end up at the same temperature but also end up blended with each other. • Is the blending necessary for reaching thermal equilibrium? • The blending points out to another type of equilibrium. • Diffusive equilibrium. • The molecules of each substance are free to move around but no tendency to move one way or another. • There is another type of equilibrium -Mechanical Equili- brium. Large scale motions can take place but no longer. • For each type of equilibrium, there is a quantity that is exchanged between the two systems.

• Exchanges Quantity Type of equilibrium

• Energy Thermal • Volume Mechanical • Particles Diffusive

• We refine further the idea of Temperature

• Temperature is a measure of the tendency of an object to absorb or lose energy spontaneously- energy going from a body at a higher temperature to a body at a lower temperature.

• Are we done defining temperature? Later we will see what temperature really is! • Most thermometers depend on the thermal expansion of materials. Other properties like electrical resistance, change of pressure can also be used. • Mercury thermometers measure the volume of a fixed amount of Hg when it expands due to heat. • How to get a numerical value for temperature?

• Pick two reference points, freezing and boiling points of water, assign them arbitrary values 0 and 100, make 100 equal divisions and declare we have a Celsius scale. • Of all the thermometers, the one based on expansion of a gas is very interesting. Why? • We can extrapolate the scale to very low temperatures . So what? • For any low density gas at constant pressure, the volume goes to zero at about -273o C. Alternatively, when V is constant, p goes to zero.

• This special temperature is called absolute zero and defines the zero point of the absolute scale. • This scale is named after William Thomson (Lord Kelvin) as the kelvin scale. • Many of the equations of thermodynamics are correct only when the temperature is measured in kelvin. How about Celsius? • Temperature is what you measure with a thermometer!

• Temperature is the thing that is the same for two objects after they have been in contact long enough

• Temperature is a measure of the tendency of an object to absorb or lose energy spontaneously- energy going from a body at a higher temperature to a body at a lower temperature. • Basic Concepts and Defintions

• System and Surroundings: A system is part of the physical world of interest. What is not the system is environment or surroundings.

• We differentiate three distinct type of systems • 1. An isolated system totally uninfluenced by the surroundings. No exchange of matter or energy with the surroundings. • 2. A closed system: Energy but not matter can exchange with the surroundings • 3. An open system: Both energy and matter can exchange. . • Isolated systems are often referred to as bodies.

• We first consider isolated and closed systems; later generalize theorems to open systems. • State Variable and thermodynamic properties • For a complete description of a macroscopic body, identity of the substance alone is not enough. The state of the system must be specified. • The state is completely specified by the values of the thermodynamic variable, p, V, T, E, n etc. • Thermodynamic properties are properties that do not depend on the rate at which somethings happen. • Electric current and thermal conductivity: are they thermodynamic properties? • No! Because they are rates! So What? For this , we have to understand state variables. • State Variables: 1.Are fully deter- mined by the values at present and do not depend on the history of the system. • 2. Not all state variable have to be specified to define the state of the system. Why? • Because the variables are interdependent and only a small number can be varied independently. • The small number of variables that can be varied independently are called the independent variable; the others are dependent variables.. • The number of independent variables to describe a macroscopic system thermodynamically is very • small, about half a dozen. • The Zeroeth Law-Temperature (Maxwell, Fowler, Sommerfeld). If two bodies are in thermal equilibrium with a third, they are in thermal equilibrium with each other. The formulation of this law has a long history. “The concept of temperature. As a natural generalization of experience we introduce the postulate: if two assemblies are each in thermal equilibrium with a third assembly, they are in thermal equilibrium with each other.

From this it may be shown to follow that the condition for thermal equilibrium between several assemblies is the equality of a certain single-valued function of the thermodynamic states of the assemblies, which may be called the temperature t, any one of the assemblies being used as a ‘thermometer’ reading the temperature t on a suitable scale. This postulate of the ‘existence of temperature’ could with advantage be known as the zeroth law of thermodynamics.”

Fowler and Guggenheim

• What is thermal equilibrium?

• Thermal equilibrium refers to systems in thermal contact that do not change with time. • Thermal contact refers to systems in contact via a diathermal wall (is onein which the state of the system can be changed by means other than moving the boundary. • How to get this single-valued function of the thermodynamic states? • Consider three systems A, B and C.

Let PA, PB ,PC, VA, VB, and VC be their mechanical variables. P = pressure, V = molar volume. • If A and C are in thermal contact across a rigid wall and in equilibrium, not all four variables are independent. • The variables are connected by an equation of state. This can be

written as (PA, VA ; PC, VC ) = 0 or PC = Φ1 (VA , VC ; PA ). • Similarly, when B and C are in

equilibrium, as (PB, VB ; PC, VC ) = 0 or PC = Φ2 (VB , VC ; PB ). • By Zeroeth law, A and B are in thermal equilibrium. This requires that

(PA, VA ; PB, VB ) = 0 Eqn (1) implying that PA, VA , PB and VB are interdependent.

• We also know that

• PC = Φ1 (VA , VC ; PA ) • = Φ2 (VB , VC ; PB ) implying a functional relationship exits between PA, PB , VA, VB, and VC Implying (PA, VA ; PB, VB; VC ) = 0. (Eqn 2.) How to reconcile the two functions and

, one implying that PA, VA ; PB, VB; are interdependent but not dependent on

VC and the other that they are dependent on VC ? This is possible if the functions Φ1 and Φ2 are of this form:

Φ1 (VA , VC ; PA ) = θ1 (VA , PA )ε (VC ) + η (VC )