TOTAL THERMODYNAMICS IN ONE ARTICLE

Internal Energy

Internal Energy U

Repetition being the apotheoses of pedagogy therefore, heat is the workless transfer of energy.

The System and Surroundings

Before we can start with the first law, it is a good idea to be clear on two important in thermodynamics: the system and the surroundings. The system is the region of the universe under study while the surroundings include everything else in the universe except the system

Figure 1. The system and surroundings

Energy is also used to mean the potential to do work; the greater the ability of something to change the things around it, the more energy it has. Mathematically, the first law is expressed:

ðU=dQ+ðW.

Where dQ is the change in the quantity of heat added to the system, ðU is the change in the internal energy and ðW is the work done on or by the system. The ð sign is used to indicate inexact differentials, meaning that the values of ðU and ðW depend on the path taken.

The sign convention here is that if dU is positive the amount of internal energy increases. This means that dQ stands for the heat energy put into the system and dW for the work done on the system. This is known as the ?physicists? convention?.

Work

	W>0	Work is done on the system by the surroundings
	W<0	Work is done by the system on the surroundings

Heat

ðQ>0	Heat is added to the system from the surroundings
ðQ<0	Heat is released by the system to the surroundings

Zeroth Law

The zeroth law is a consequence of thermal equilibrium and allows us to conclude that temperature is a well-defined physical quantity. The zeroth law of thermodynamics states:

If a body A and a body B are both in equilibrium with each other; then a body C which is in thermal equilibrium with body B will also be in equilibrium with body Aand the temperature of body C is equal to the temperature of body A.

It is the zeroth law, because it preceeds the first and second laws of thermodynamics and is also a tacit assumption in both laws.

We use the zeroth law when we wish to compare the temperatures of two objects, A and B. We can do this by using a thermometer, C and placing it again object A it reaches thermal equilibrium with object A and measure the temperature of A. Placing the thermometer against object B until thermal equilibrium is reached we measure the temperature of object B. If they are the same temperature then they will be in thermal equilibrium with each other.

Entropy

Introduction

The concept of entropy is particularly abstract and by the same token difficult to present. Yet some scientists consider it intuitively; they need only refer mentally to actual states such as disorder, waste, and the loss of time or information. But how can degraded energy, or its hierarchy, or the process of degradation be truly represented?

Entropy is a state variable which measure the disorder of a system or the amount of energy available to be used. Mathematically, in thermodynamics entropy is given by S=Q/T where S is the entropy, Q is the internal energy and T is the temperature.

The Second Law of Thermodynamics

Introduction

No process is possible whose only result is the abstraction of heat from a system and the performance of an equivalent amount of work. There is no such thing as a 100% efficient system; it is impossible to turn all the heat absorbed into mechanical work, and that which is not used in mechanical work generates entropy. This defines entropy as a mathematical construct which only remains constant in a perfectly efficient (but hypothetical) closed thermodynamic cycle. The second law of thermodynamics defines the heat absorbed thus:

dQ&gte;TdS

where dS is the change in entropy, and is therefore given by:

dS> dQ/T;

The change in entropy increases by this amount for a reversible process, and by a larger amount for an irreversible process (a complete and irretrievable departure from equilibrium, such as diffusion, explosions, etc., all of which are one-way process unless there is a significant amount of external intervention). This is because in the latter case the greater change in entropy is due to the nature of irreversibility, e.g., bonds being broken generating a dissociation energy. ?S is always greater than zero with respect to the whole universe, so entropy itself of such an isolated system is always increasing. Thus for the change in internal energy we have:

dU = T dS - P dV

The state variables (P, V, T) define a unique state for a system.

The second law of thermodynamics places a direction of flow on heat energy. The energy flows from a hotter regions to colder regions to maintain thermal equilibrium.

A summary of the second law was given by C.P. Snow:

1. You cannot win. (energy and matter are conserved, so you cannot get something for nothing.)
2. You cannot break even (there is always an increase in entropy so you cannot even return to the same state.)
3. You cannot leave the game. (there is no way to escape rules 1 and 2 because it is impossible to reach absolute zero (see third law)

Latent Heat

When a substance changes phase, that is it goes from either a solid to a liquid or liquid to gas, the energy, it requires energy to do so. The potential energy stored in the interatomics forces between molecules needs to be overcome by the kinetic energy the motion of the particles before the substance can change phase.

If we measure the temperature of the substance which is initially solid as we heat it we produce a graph like Figure 1.

Figure 1. Temperature change with time. Phase changes are indicated by flat regions where heat energy used to overcome attractive forces between molecules

Starting a point A, the substance is in its solid phase, heating it brings the temperature up to its melting point but the material is still a solid at point B. As it is heated further, the energy from the heat source goes into breaking the bonds holding the atoms in place. This takes place from B to C. At point C all of the solid phase has been transformed into the liquid phase. Once again, as energy is added the energy goes into the kinetic energy of the particles raising the temperature, (C to D). At point D the temperature has reached its boiling point but it is still in the liquid phase. From points D to E thermal energy is overcoming the bonds and the particles have enough kinetic energy to escape from the liquid. The substance is entering the gas phase. Beyond E, further heating under pressure can raise the temperature still further is how a pressure cooker works.

Latent Heat of Fusion and Vaporisation

The energy required to change the phase of a substance is known as a latent heat. The word latent means hidden. When the phase change is from solid to liquid we must use the latent heat of fusion, and when the phase change is from liquid to a gas, we must use the latent heat of vaporisation.

The energy require is Q= m L, where m is the mass of the substance and L is the specific latent heat of fusion or vaporisation which measures the heat energy to change 1 kg of a solid into a liquid.

Specific Heat Capacity

Energy is required to raise the temperature of an object. How much energy is required depends on what the object is made of. If we are going to compare how much energy is required to heat an object we must consider we are comparing like-with-like. The specific heat capacity gives us the energy required to raise the temperature of unit mass by one-degree Centigrade.

The word specific in physics has a specific definition, it means a mass of 1kg.

To measure the energy required we use a source of heat, either electrical or chemical. The specific heat capacity is then the energy input = mass x specific heat capacity x the change in temperature.

Mathematically, Q=mc??, where m is the mass of the object being heated, c is the specific heat capacity of the material the object is made from and ?? is temperature difference between the final and initial temperatures in K or °C.

Thermal Equilibrium

A body is in thermal equilibrium when there is no energy transfer between the body and its surroundings.

Phase Diagrams

The phases of the material can be recorded for many different pressures and temperatures. Plotting the phases, whether the material is solid, liquid or gas for many different pressures and temperatures we can build up a phase diagram for the substance. As shown in Figure 2.

Figure 2. Phase Diagram for water

The phase diagram shows that at the interfaces between solid and liquid, liquid and gas and solid and gas it is possible for more than one phase to exist in equilibrium. The point at which all three phases come to gether is the triple point and represents the temperature and pressure for which all three states of matter can exist. For water this is, 273.16 K at 611.2 Pa. The other labeled point on the diagram is called the critical point, also called critical state. At this point the liquid and gaseous phases of a pure stable substance become identical.

State Variables

In thermodynamics we define the state of a substance in terms of the various properties we can attribute to it.

Temperature T	Pressure P
Volume V	Entropy S
Enthalpy H	Internal Energy Q
Mass m	Density ?

Reversible Processes

A process is said to be reversible when the successive states of the process are infinitesimally close to equilibrium States. i.e. the process is quasi-equilibrium.

With a reversible process it is possible to restore the system to its original state without needing an external agent or changing its surroundings.Reversible processes are an abstraction that aids the analysis of real processes.

A reversible process is a standard of comparison for an actual system. Truly reversible thermal processes would require an infinite amount of time for completion.

If a system is in equilibrium it state variable do not change with time

Quasi Equilibrium

A process is called a quasi-equilibrium process if the intermediate steps in the process are all infinitesimaly close to equilibrium. In this way we can characterize the intermediate states of the process using state variables.

When a process is quasi-equilibrium we can plot a graph of P against V the path of the process. Areas under the graph then represent the work since all the variable used to characterize the substance's intermediate states have well defined values.

Most of the process you will encounter will be quasi-equilibrium processes

Irreversible Processes

In practice, all Natural processes are Irreversibl processes. The path of an irreversible process is indeterminate and cannot be drawn on a thermodynamic diagram. (We use a hashed line to indicate the path because the intermediate states are in non-equilibrium.)

The Entropy of the universe always increases during an irreversible process. It is always possible to restore an irreversible process to its original state by a reversible process, but the Entropy of the universe can never be restored. An irreversible process always requires an external agent to restore it to its original state.

Examples of Irreversible Processes include:

Friction
Heat Flow
Unrestrained Expansion
Melting/Boiling
Mixing
Inelastic Deformation
Chemical Reactions
Current Flow
Your House Getting Dirty

The Third Law of Thermodynamics

A postulate related to but independent of the second law is that it is impossible to cool a body to absolute zero by any finite process. Although one can approach absolute zero as closely as one desires, one cannot actually reach this limit. The third law of thermodynamics, formulated by Walter Nernst and also known as the Nernst heat theorem, states that if one could reach absolute zero, all bodies would have the same entropy. In other words, a body at absolute zero could exist in only one possible state, which would possess a definite energy, called the zero-point energy. This state is defined as having zero entropy.

As the entropy of a substance approaches zero, it temperature approaches absolute zero.

Heat Engines

A heat engine is any machine which converts heat into useful work for example, a steam engine or a car engine. Real heat engines are complex and there are many ways of converting heat energy into useful work. We can abstract and generalise the workings of any heat engine into three parts:

• The Hot Resevoir - heat energy is created by some process such as combustion of a fuel to provide the heat energy.
• The working body - converts the heat energy into work. In real heat engines, the conversion process is never 100% efficient, so the work output is always less than the heat energy supplied. However we frequently idealise and assume reversibility.
• The cold resevoir - the energy that cannot be turned into work is dumped and goes to heat up the cold resevoir. In practice, the cold resevoir is usually the atmosphere. We also assume that the temperature of the cold resevoir does not increase, it has an infinite heat capacity.

Figure 1. Schematic diagram of a heat engine.

Assume that a heat engine starts with a certain internal energy U, intakes ?Q_i heat from a heat source at temperature T_i , does work ?W , and exhausts heat ?Q_f into a the cooler heat reservoir with temperature T_f. With a typical heat engine, we only want to use the heat intake, not the internal energy of the engine, to do work, so ?U=0. The first law of thermodynamics tells us:

?U=0 = ?Q_i - ?Q_f - ?W

To determine how effectively an engine turns heat into work, we define the efficiency, ?, as the ratio of work done to heat input:

? = ?W/?Q_i = (?Q_i-?Q_f)/?Q_i

= 1 - ?Q_f/?Q_i

Because the engine is doing work, we know that ?W >0, so we can conclude that ?Q > 0. Both and are positive, so the efficiency is always between 0 and 1:

Efficiency is usually expressed as a percentage rather than in decimal form. That the efficiency of a heat engine can never be 100% is a consequence of the Second Law of Thermodynamics. If there were a 100% efficient machine, it would be possible to create perpetual motion: a machine could do work upon itself without ever slowing down.

Enthalphy

The enthalpy, H is the heat content of a system. It can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. The total enthalpy cannot be measured directly so the enthalpy change of a system is measured.

Mathematically, the enthalpy is defined as

?H = ?U + P dV

Gibbs Free Energy

The Gibbs free energy is constant if the temperature and pressure are constant. These are the conditions under which phase transitions (melting, boiling) take place, and are also relevant to chemical equilibrium.

The Gibbs Free Energy is given by:

G = U + P V - T S

Where U is the internal energy, P the pressure, V the volume, T the temperature in Kelvin, S the entropy.

The Gibbs Free Energy can be though of as the maximum energy available from a chemical reaction.

The Ideal Gas Equation

The relationships between volumes of gases under constant temperature and pressure were established by Boyle, Charles.

Boyles law was first published in 1662. It states

The pressure times the volume of a gas is equal to some constant k. PV= k.

Charles/Gay-Lussac Law, states that

At constant pressure, the volume of a given mass of an ideal gas increases or decreases by the same factor as its temperature (in kelvin) increases or decreases.

P₁/T₁ = P₂/T₂

Both of these laws can be combined to give an overall equation which describes the behaviour of gases.

P₁V₁/T₁ = P₂V₂/T₂

Avogadro's Law, states that the volume V is related to the number of moles of gas, n by

V/n = k

P₁V₁/T₁n = P₂V₂/T₂n = k, where k is a constant.

Which is more familiar as

PV= nRT

The constant k has been replaced by R the molar gas constant.

Where P is the pressure, V is the volume, n is number of moles of gas present, R is the molar gas constant has a value very close to 8.31 m² kg s^-2 K^-1 mol^-1 and T is the temperature in Kelvin.

The PV diagram

We can plot the path of the pressure against the volume, if we consider the change in the state variables to be quasi-static.

For a constant temperature the pressure against volume curve looks like

Isothermal Process

Example1: Boiling of water in the open air. In general most isobaric phase changes are isothermal. In this example the system does work as the steam-produced pushes against the atmosphere as it expands. Neither the heat Q , the work W, or the change in internal energy DU are zero. In this case Q = mLv since the water changes phase. Example 2: In general for an Ideal gas U is only a function of the temperature so that DU is always equal to zero for an isothermal process.Since DU = 0 then W = Q from the First Law. What has to happen for this process to be isothermal is that the gas in a cylinder is compressed slowly enough that heat flows out of the gas at the same rate at which is being done on the gas. The ideal gas law can be used to determine the work done W = PV ln(Vf/Vo) which is also the equation for Q. Note that P1V1 = P2V2 = nRT, the ideal gas law for an isothernal process.

Adiabatic Process

Example Compression of a Gas in an Insulated Cylinder. In this case any change in the internal energy of the gas is due to work done on it or by it, DU = W. Normally if DU changes the temperature of a system will change. Any temperature rise or fall is due to the work done or by the gas alone and not due to heat flowing into or out of the system since Q = 0. If a process is carried out fast enough the heat flow will be small and the process can be approximate as being adiabatic. This happen because heat flow is in general a slow process. Observe that we did not say that Q is constant because it not a state variable. Q represent an energy transfer not the heat energy of the system. In addition to the ideal gas law PV = NkT, the quantity PVg is constant for an ideal gas where g = cP/cV, the ratio of molar specific heats. For an ideal gas the work W = (P1V1 - P2V2)/(g - 1)

Isobaric Process

Example

Gas Heated in a Cylinder fitted with a movable frictionless piston. The pressure the atmosphere and the pressure due to the weight of the piston remains constant as the gas heats up and expands. First Law Implications: ?U = Q - W Unlike some of the other processes below neither the heat Q , the work W, or the change in internal energy ?U are necessarily zero in a constant pressure process. For an ideal gas, constant pressure work is easily determined, W = Ú PdV = P ?V Part of the heat that flows into the system causes the temperature to rise, Q = n cp DT = m Cp DT, the rest goes into work.

Work

When a thermodynamic system does work, it uses energy in the form of heat added to the system. The general definition of work is that of a force times distance.

Consider a freely moving piston in a cyclinder which is closed at one end. Inside the cyclinder, there is a volume of gas inside the cylinder which is at a certain temperature T, pressure P and takes up a volume V. If heat is added to the system, the temperature is raised, the molecules in the gas gain kinetic energy and hit the walls of the cylinder with increased velocity. The force on the walls of the cylinder increase. The cylinder walls cannot move, but the piston, which is free to move, experiences a reaction force F=P A, where P is the pressure and A is the area of the piston. The piston moves a distance ?x. The work done dW=?P A ?x. If A is constant, A ?x is the change in volume. For a constant force, W= PdV.Therefore the work done is P dV.

When the volume of a gas increases, work is done by the gas.

When the volume of a gas decreases, work is done on the gas by an external force.

Quasi-Static Process

If we consider the system to change slowly over time, we can effectively, consider each state to be considered reversible

The work is dependent on the path over which the process takes place.

The work done in going from 1 to 2 via a is

W= 2p₀V₀

but it is

W= p₀V₀

In going from 1 to 2 via b.

The above result shows that if the temperature of a gas is increased at constant volume, no work is done.

However, if the temperature is increased and the gas is allowed to expand, work will be done. In this case, extra energy will have to be supplied to do this work.

For this reason, gases are said to have two principal specific (or molar) heat capacities:

i) the specific (or molar) heat capacity at constant volume, cv ii) the specific (or molar) heat capacity at constant pressure, cp It should be clear that cp > cv and that the difference between them is given by

cp - cv = pDV

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