Thermodynamics

Contents

Terminology

It is essential to define the terminology before learning more about the subject:


• Heat - Heat (Q) is a form of energy transfer associated with random motion of the microscopic particles.
• Work - Work (W) is the organized form of energy transfer associated with the motion of microscopic particles as a whole (in a certain direction), e.g., the expanding gas that propels a piston.
• Internal Energy - The internal energy (U) of a system is the total energy due to the motion of molecules, plus the rotation, and vibration of atoms within molecules. Heat and work are two methods of adding energy to or subtracting energy from a system. They represent energy in transit and are the terms used when energy is moving. Once the transfer of energy is over, the system is said to have undergone a change in internal energy dU. Thus, in terms of the amount of heat dQ and work dW:
  
  dU = dQ + dW ---------- (1)
  
  where dQ and dW are positive for energy transfer from the surroundings to the system, and negative for energy transfer from the system to the surroundings. If the process of energy transfer is broken down into finer details, e.g., change in disorder (dS), volume expansion/contraction (dV), and adding a new species of particles (dN), then the change in internal energy can be expressed as:
  
  dU = T dS - p dV + dN ---------- (2)
  
  
• Temperature - Temperature (T) is related to the amount of internal energy in a system. As more heat or work is added the temperature rises, similarly a decrease in temperature corresponds to a loss of heat or work performed from the system. Temperature is an intrinsic property of a system, meaning that it does not depend on the system size or the amount of material in the system. Other intrinsic properties include pressure and density. The internal energy (U) is related to the temperature (T) by the formula:
  
  U= (3nR/2) T ---------- (3)
  
  where R = 8.314x107 erg/K^o-mole is called the gas constant.



• Pressure - Pressure (p) is the force normal to the surface of area upon which it exerts. Microscopically, it is the transfer of momenta from the particles that produces the force on the surface.
• Volume - Volume (V) is referred to the three dimensional space occupied by the system.
• Particle Number - Particle number (N) is the number of a particular constituents in a system.
• Avogadro's Number - Avogadro's number (N₀) is 6.023 x 10²³. One mole is defined as the unit that contains that many number of particles such as atoms, molecules, or ions.
• Number of Moles - Number of moles (n) is the number of particles in the unit of a mole, i.e., n = N / N₀.
• Density - Density () is defined as mass per unit volume.
• Entropy - Entropy (S) is a measure of disorder in the system. Mathematically, the change of entropy dS is related to the amount of heat transfer dQ by the formula:
  
  dS = dQ / T    or    dQ = T dS ---------- (4)
  
  
• Chemical Potential - The chemical potential () of a thermodynamic system is the change in the energy of the system when a different kind of constituent particle is introduced, with the entropy and volume held fixed.

Systems

There are three kinds of system depending on the kinds of exchanges taking place between a system and its environment:


1. Isolated System - It does not exchange heat, matter or work with their environment. An example of an isolated system would be an insulated container, such as an insulated gas cylinder. In reality, a system can never be absolutely isolated from its environment, because there is always at least some slight coupling, even if only via minimal gravitational attraction.
2. Closed System - It exchanges energy (heat and work) but not matter with the environment. A greenhouse is an example of a closed system exchanging heat but not work with its environment.
3. Open System - It exchanges energy (heat and work) and matter with the environment. A boundary allowing matter exchange is called permeable. The ocean would be an example of an open system.

States

The Four Laws of Thermodynamics

• Zeroth law - It is the definition of thermodynamic equilibrium. When two systems are put in contact with each other, energy and/or matter will be exchanged between them unless they are in thermodynamic equilibrium. In other word, two systems are in thermodynamic equilibrium with each other if they stay the same after being put in contact.
  
  The original zeroth law is stated as If A and B are in thermodynamic equilibrium, and B and C are in thermodynamic equilibrium, then A and C are also in thermodynamic equilibrium.
  
  Thermodynamic equilibrium includes thermal equilibrium (associated to heat exchange and parameterized by temperature), mechanical equilibrium (associated to work exchange and parameterized generalized forces such as pressure), and chemical equilibrium (associated to matter exchange and parameterized by chemical potential).



• 1st Law - This is the law of energy conservation. It is stated alternatively in many forms as follows:
  
  The work exchanged in an adiabatic process depends only on the initial and the final state and not on the details of the process.
  or
  The heat flowing into a system equals the increase in internal energy of the system minus the work done by the system.
  or
  Energy cannot be created, or destroyed, only modified in form.
  
  The second statement can be expressed mathematically in the form of Eq.(1) with negative W representing work done by the system. The adiabatic process in the first statement refers to a system with no heat transfer, i.e., Q = 0.



• 2nd Law - It can be stated in many ways, the most popular of which is:
  
  It is impossible to obtain a process such that the unique effect is the subtraction of a positive heat from a reservoir and the production of a positive work.
  or
  A system operating in a cycle cannot produce a positive heat flow from a colder body to a hotter body.
  
  The first statement is to exclude the un-realistic situations such as to drive a steamship across the ocean by extracting heat from the water, or to run a power plant by extracting heat from the surrounding air. The second statement expresses the impossibility of running refrigeration without work. Another form of the 2nd law states:
  
  The entropy of a closed system tends to remain constant or to increase.
  
  It is in this form that the arrow of time is defined.



• 3rd Law: This law explains why it is so hard to cool something to absolute zero:
  
  All processes cease as temperature approaches zero.
  
  This statement is expressed mathematically by Eq.(4), which shows that as the temperature T approaches zero the amount of heat extracted from the system also diminishes to zero. Thus, even using laser cooling would not be able to attain a temperature of absolute zero.

Thermodynamic Process

• Adiabatic Process - This is a process that takes place in such a manner that no heat enters or leaves a system. Such change may be accomplished either by surrounding the system with a thick layer of heat insulating material or by performing the process quickly. The flow of heat is a fairly slow process; so that any process performed quickly enough will be practically adiabatic. The compression and expansion phases of a gasoline engine is an example of an approximately adiabatic process.
• Isochoric Process - If a system undergoes a change in which the volume remains constant, the process is called isochoric. The explosion of gasoline vapor and air in a gasoline engine may be treated as though it were an isochorie addition of heat
• Isobaric Process - A process taking place at constant pressure is call an isobaric process. When water enters the boiler of a steam engine and is heated to its boiling point, vaporized, and then the steam is superheated, all these processes take place isobarically.
• Isothermal Process - Isothermal process changes the system slowly so that there is enough time for heat flow to maintain a constant temperature. Slow change is a reversible process, because at any instant the system is in its most probable configuration. In general, a process will be reversible if:
  
  1. it is performed quasistatically (slowly);
  2. it is not accompanied by dissipative effects, such as turbulence, friction, or electrical resistance.
• Isentropic Process - If the slow change is accomplished in an insulated container, there is no heat flow. According to
  Eq.(4) there is also no change in entropy. Thus, a reversible adiabatic process is isentropic.

Work and Engines

		The most efficient heat engine cycle is the Carnot cycle, consisting of two isothermal processes and two adiabatic processes (see Figure 02). The Carnot cycle can be thought of as the most efficient heat engine cycle allowed by physical laws. When the second law of thermodynamics states that not all the supplied heat in a heat engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat which can be so used. In order to approach the Carnot efficiency, the processes involved in the heat engine cycle
Figure 01 Heat Engine [view large image]	Figure 02 Carnot Engine Cycle [view large image]	must be reversible and involve no change in entropy. This means that the Carnot cycle is an idealization, since no real engine processes are reversible and all real physical processes involve some increase in entropy.

			The p-V diagrams for the more realistic cases are shown in Figure 03, 04, and 05 for the gasoline, diesel, and steam engines respectively. While the gasoline and diesel engines operate at about 50% efficiency, the steam engine runs at only about 30%. A brief description of the processes can be found in each of the diagram.
Figure 03 Gasoline Engine [view large image]	Figure 04 Diesel Engine [view large image]	Figure 05 Steam Engine [view large image]

Connection to the Microscopic View

---------- (7)

This is known as the Boltzmann equation. It is very useful as mathematic tool in treating the process of fluid flow. By expanding the distribution funciton in terms of the power of the velocity - v0, v1, and v2, the equilibrium distribution in the form of Maxwell-Boltzmann distribution, the continuity equation, and the Navier-Stokes equations in fluid dynamics can be derived directly from Eq.(7). Analytical solutions of the Boltzmann equation are possible only under very restrictive assumptions. Direct numerical methods for computer simulation have been limited by the complexity of the equation, which in the complete 3-D time-dependent form requires seven independent variables for time, space and velocity. A 2-dimensional animation of a flow process is presented by clicking Figure 06. It shows the development of a clump of gas molecules initially

Figure 06 Boltzmann Equation Simulation [view animation]

released from the left. The particles flow to the right, reflected by the wall at the other end, then established an equilibrium configuration after some 4000 collisions between the particles.

---------- (8)

-t/

-t/

---------- (10)

---------- (11)

temperature, most of the bosons occupy the same state with E ~ 0. This is the Bose-Einstein condensate first discovered in 1995. Another example of Bose-Einstein distribution is the black-body radiation. In Fermi-Dirac distribution, the normalization constant A can be re-defined as A = e-Ef, where Ef is known as the Fermi energy, which has a value of a few ev for the electron gas in many metals. Note that f (E) = 1/2 at E = Ef for all temperatures. At low temperature most of the low energy states with E < Ef are filled. At high temperature with kT >> (E - Ef),

Figure 07 Distribution Functions [view large image]

the distribution function becomes f (E) ~ (1/2) (1 - (E - Ef) / 2kT). Thus in this case, the energy states with E < Ef are more than half-filler; while for E > Ef they are less than half-filled.

In classical statistic, the velocity distribution of the ideal gas is given by the Maxwell distribution as shown in Figure 08. A relationship between the root-mean-square velocity vrms and the temperature T can be derived from such distribution function: m vrms2 = 3 k T    or   M vrms2 = 3 R T ---------- (12) where m denotes the mass of the molecule, M = mN0 is the molecular weight/mole, N0 is the Avogadro's number, and k = R / N0 = 1.38x10-16 erg/Ko is the Boltzmann constant .

Figure 08 Maxwell Distribution [view large image]

The formula in Eq.(12) provides a link between the microscopic root-mean-square velocity vrms of the particles and the macroscopic property T.

The mean free path (Figure 09) can be expressed mathematically as: l = 1 / nA = (l1 + l2 + l3 + ... + lN) / N---------- (13) where n is the number density, A is the collision cross section, li is the path length between collisions, i.e., length of the free path, and N is the total number of collisions. The concept of mean free path may be visualized by thinking of a man shooting a rifle aimlessly into a forest. Most of the bullets will hit trees, but some bullets will travel much farther than others. The

Figure 09 Mean Free Path [view large image]

average distance traveled by the bullets will depend inversely on both the denseness of the woods and the size of the trees.