In physics, thermodynamics (from the Greek Î¸ÎÏÎ¼Î· therme, meaning "heat" and Î´ÏÎ½Î±Î¼Î¹Ï‚, dynamis, meaning "power") is the study of energy conversion between heat and mechanical work, and subsequently the macroscopic variables such as temperature, volume and pressure. Its progenitor, based on statistical predictions of the collective motion of particles from their microscopic behavior, is the field of statistical thermodynamics (or statistical mechanics), a branch of statistical physics.
Thermodynamics is the science which relates dynamics of fluids with thermal and energy, thermodynamics deals with heat, work, and power. In this report a set of statements and formulas are described and explained.
State of a System, 0th law of thermodynamics:
The zeroth law of thermodynamics states that when two bodies have equality of temperature with a third body, they in turn equality of temperature with each other [Gordon J. Van Wylen].
If A, B, and C are systems or bodies, we said that the bodies or the systems are in thermal equilibrium or constant temperature, A and B in thermal equilibrium and B and C are in thermal equilibrium also.
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if T (A) = T(B)
and T (B) = T(C)
then T (A) = T(C).
Figure (1): Thermal equilibrium between two bodies.
Work, Heat, 1th law of thermodynamics:
The first law of thermodynamics states that during a cycle a system (control mass) undergoes, the cyclic integral of the heat is proportional to the cyclic integral of the work [Gordon J. Van Wylen].
In another words the conservation of energy states that the change in the internal energy of any closed system equal the heat added to the system minus the work done by the system. the following equation shows that:
Consider piston cylinder system with water inside the cylinder, state (1) as shown in figure (2) below shows the initial state of the system (water has internal energy) and it is in equilibrium state, when an external load applied to the piston the system transferred to state (2) and work and heat transferred into and from the system to reach to the second equilibrium position (state 2).
Figure (2): Application of the first law of thermodynamics.
Internal Energy, Expansion Work:
The internal energy is a thermodynamic property; also it can be defined as the amount of random energy included in certain amount of the mater due to the internal movement of atoms. Also it is extensive property because it depends on the mass of the system.
The amount of internal energy of any material as thermodynamic property depends on the mass of the body and it specific heat capacity, for example if we increase the temperature of metal its internal energy increased based on the temperature difference, also metals have high specific heat capacity than liquids.
Figure (3): Comparison between metal and ice based on the internal energy.
The enthalpy is defined as the heat transfer during the process which is given in the terms of the change in internal energy, pressure and volume [Gordon J. Van Wylen]. The following equation shows the main parameters of enthalpy.
The thermodynamic potential H was introduced by the Dutch physicist Kamerlingh Onnes in early 20th century in the following form:
Where E represents the energy of the system. In the absence of an external field, the enthalpy may be defined, as it is generally known, by:
where (all units given in SI)
H is the enthalpy (in joules),
U is the internal energy (in joules),
p is the pressure of the system, (in pascals), and
V is the volume, (in cubic meters).
Form pV (sometimes called "flow work") is motivated by the following example of an isobaric process. Gas producing heat (by, for example, a chemical reaction) in a cylinder pushes a piston, maintaining constant pressure p and adding to its thermal energy. The force is calculated from the area A of the piston and definition of pressure p = F/A: the force is F = pA. By definition, work W done is W = Fx, where x is the distance traversed. Combining gives W = pAx, and the product Ax is the volume traversed by the piston: Ax = V. Thus, the work done by the gas is W = pV, where p is a constant pressure and V the expansion of volume. Including this term allows the discussion of energy changes when not only temperature, but also volume or pressure are changed. The enthalpy change can be defined Î”H = Î”U + W = Î”U + Î”(pV), where Î”U is the thermal energy due to heating of the gas during the expansion, and W the work done on the piston.
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Joule-Thomson experiment is used to determine the carbon dioxide coefficient. And comparing the experimental value with the calculated value. Figure (3) shows the experimental apparatus of Joule-Thomson experiment.
Figure (3): Joule-Thomson experimental setup (Taylor).
The fluid allowed flowing steadily from a high pressure to low pressure through a porous plug inserted in a pipe. At steady conditions the pipe is insulated from any heat loss to surrounding, the flow velocity should be low so the differences in kinetic energy between the upstream and the downstream are negligible. Measurements' of temperature and pressure up stream and downstream the media should be taken (G.F.C. Rogers).
Plotting curves for both heating and cooling process for pressure and temperature of the gas, the plotted curves are shown in figure (4).
Figure (4): Isenthalpic curves and the enthalpy inversion curve (Taylor).
Adiabatic means the process during which the heat is prevented from crossing the boundary of the system (G.F.C. Rogers). The system is thermally insulated from the surrounding conditions, so for adiabatic process the first law of thermodynamics is reduced to the change in internal energy equals the work done by the system or on the system.
Figure (5): Adiabatic process in P-V diagram (G.F.C. Rogers).
What is Thermochemistry:
Thermochemistry is the study of energy produced or absorbed in chemical reactions and any physical transformation such as melting or boiling. Thermochemistry, generally, is concerned with the energy exchange accompanying transformations, such as mixing, phase transitions, chemical reactions, and including calculations of such quantities as the heat capacity, heat of combustion, heat of formation, enthalpy, and free energy (E.H. Cole). Thermochemistry rests on two generalizations. Stated in modern terms, they are as follows:
Lavoisier and Laplace's law (1780): The energy change accompanying any transformation is equal and opposite of energy change accompanying the reverse process.
Hess's law (1840): The energy change accompanying any transformation is the same whether the process occurs in one step or many.
Figure (6): Energy movement (www.howstuffworks.com).
What is Calorimetry:
The word calorimetry was derived from the lateen word calor which means heat and Greek word metry which means measure; it is the science of measuring the amount of heat. To measure the energy produced from certain fuel or matter calorimeter is used. Calorimeter is a device consists of barrel filled with water and a bomb filled with fuel (oil fuel or coal) also and electric circuit is used to produce electrical signal to burn the discharge inside the bomb, after that the heat transfers to the water inside the calorimeter, by measuring the initial and final water temperature and knowing the water amount in the calorimeter, the amount of heat produce from the fuel discharge can be estimated. The figure below shows the calorimeter.
Figure (7): Calorimeter (E.H. Cole).
Second Law of Thermodynamics:
The second law of thermodynamics is the law of heat and power, it can be expressed as:
It is impossible to cause an engine to operate in a (thermodynamics) cycle, in which the only interactions are positive work done on the surroundings and heat transfer from a system which remains at constant temperature (E.H. Cole).
Figure (8): The schematic of second law of thermodynamics (www.howstuffworks.com).
The following formula of the law has been proposed:
It is impossible to construct a heat-engine cycle which will produce only the effect of rising a weight (net work or shaft work) if heat is exchanged with a single thermal reservoir (Max Planck), and heat can not of itself flow from a colder to a hotter system (Rudolf Clausius).
Said Carnot a French scientist of the early nineteenth century, he proposed a heat engine cycle based on the second law of thermodynamics. Carnot said that the work by the heat engine cycle increased by increasing the temperature differences between the hot and the cold reservoirs (Leonard). So the efficiency of Carnot cycle depends on the temperatures of the hot and cold reservoirs.
Figure (9): Caront cycle in Pressure-Volume diagram, (www.howstuffworks.com).
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The performance of heat engine cycle ix expressed as the dividend divided by the cost, the purpose of power cycle is to deliver shaft work, which is the dividend. The cost depends on the heat supply from the hot reservoir.
Third law of Thermodynamics and absolute entropy:
The Third Law of Thermodynamics is the lesser known of the three major thermodynamic laws. Together, these laws help form the foundations of modern science. The laws of thermodynamics are absolute physical laws everything in the observable universe is subject to them. Like time or gravity, nothing in the universe is exempt from these laws. In its simplest form, the Third Law of Thermodynamics relates the entropy (randomness) of matter to its absolute temperature (G.F.C Rogers).
The Third Law of Thermodynamics refers to a state known as "absolute zero." This is the bottom point on the Kelvin temperature scale. The Kelvin scale is absolute, meaning 0Â° Kelvin is mathematically the lowest possible temperature in the universe. This corresponds to about -273.15Â° Celsius, or -459.7 Fahrenheit.
In actuality, no object or system can have a temperature of zero Kelvin, because of the Second Law of Thermodynamics. The Second Law, in part, implies that heat can never spontaneously move from a colder body to a hotter body. So, as a system approaches absolute zero, it will eventually have to draw energy from whatever systems are nearby. If it draws energy, it can never obtain absolute zero. So, this state is not physically possible, but is a mathematical limit of the universe. In its shortest form, the Third Law of Thermodynamics says: "The entropy of a pure perfect crystal is zero (0) at zero Kelvin (0Â° K)." Entropy is a property of matter and energy discussed by the Second Law of Thermodynamics. The Third Law of Thermodynamics means that as the temperature of a system approaches absolute zero, its entropy approaches a constant (for pure perfect crystals, this constant is zero).
A pure perfect crystal is one in which every molecule is identical, and the molecular alignment is perfectly even throughout the substance. For non-pure crystals, or those with less-than perfect alignment, there will be some energy associated with the imperfections, so the entropy cannot become zero. The Third Law of Thermodynamics can be visualized by thinking about water. Water in gas form has molecules that can move around very freely. Water vapor has very high entropy (randomness). As the gas cools, it becomes liquid. The liquid water molecules can still move around, but not as freely. They have lost some entropy. When the water cools further, it becomes solid ice. The solid water molecules can no longer move freely, but can only vibrate within the ice crystals. The entropy is now very low. As the water is cooled more, closer and closer to absolute zero, the vibration of the molecules diminishes. If the solid water reached absolute zero, all molecular motion would stop completely. At this point, the water would have no entropy (randomness) at all.
Criteria of Equilibrium:
The state of system is determined by the molecules within the system boundaries. The equilibrium has different meanings, if we have material in solid or liquid phase we said that material is in phase equilibrium if its phase does not change. Also if the state of the material is constant we said that material in thermodynamic equilibrium (William C. Reynolds).
The macroscopic properties that can in principle be measured as a function of the thermodynamic equilibrium state and that are in some way relevant to energy are called thermodynamic equilibrium. Any conglomerate feature of all the molecules, such as their total energy, is a thermodynamic property. When the state is fixed the thermodynamics properties are fixed.
13. Helmholtz and Gibbs free energy:
The thermodynamics potentials consists of four quantities, these quantities are internal energy, the enthalpy, the Helmholtz free energy and the Gibbs free energy. So Helmholtz and Gibbs are part of thermodynamics potential.
The Helmholtz free energy depends on the internal energy, temperature, and entropy. Equation below shows the relation between internal energy, absolute temperature, and entropy in Helmholtz free energy equation.
Gibbs free energy as shown in equation below depends on internal energy, absolute temperature, entropy, absolute pressure, and the final volume.
The four thermodynamic potentials are related by offsets of the "energy from the environment" term TS and the "expansion work" term PV. A mnemonic diagram suggested by Schroeder can help you keep track of the relationships between the four thermodynamic potentials.
14. Hess's law:
Hess's law states that the energy change in any chemical or physical reaction does not depend on the path or number of steps required to complete this reaction.
Figure (10): Reaction steps with energy amount.
The Î”H for a single reaction can be calculated from the difference between the heats of formation of the products minus the heat of formation of the reactants. In mathematical terms:
15. Clausius-Clapeyron equation:
The Clausius-Clapeyron equation relates the variation of pressure with temperature along the saturated-vapor (or liquid) line to the enthalpy and volume of vaporization. This equation is useful in constructing a graphical or tabular equation of state from a minimum of experimental measurements (Williams C. Reynolds).
The clausius-Clapeyron equation allows estimating the vapor pressure at any temperature if the enthalpy of vaporization and vapor pressure at some temperatures are known,
16. Ideal Solution and Non-ideal Solution:
In chemistry, an ideal solution or ideal mixture is a solution in which the enthalpy of solution (or "enthalpy of mixing") is zero; the closer to zero the enthalpy of solution is, the more "ideal" the behavior of the solution becomes. Equivalently, an ideal mixture is one in which the activity coefficients (which measure deviation from ideality) are equal to one (Wikipedia, the free encyclopedia). A solution whose behavior does not conform to that of an ideal solution; that is, the behavior is not predictable over a wide range of concentrations and temperatures by the use of Raoult's law.
In contrast to ideal solutions, where volumes are strictly additive and mixing is always complete, the volume of a non-ideal solution is not, in general, the simple sum of the volumes of the component pure liquids and solubility is not guaranteed over the whole composition range.
Figure (11): Behavior of non ideal solutions.
17. Statistical mechanics:
Statistical mechanics or statistical thermodynamics is a mathematical tool deals with high population or data. It's related with macroscopic thermodynamic properties such as work, entropy, free energy, and heat.
Ludwig Boltzmann is the father of statistical thermodynamics; he started the work in statistical mechanics in 1870.
18. Raoult's Law /MIXTURES:
The partial vapour pressure of a component in a mixture is equal to the vapour pressure of the pure component at that temperature multiplied by its mole fraction in the mixture.
Raoult's Law only works for ideal mixtures In equation form, for a mixture of liquids A and B, this reads (http://www.chemguide.co.uk/physical/phaseeqia/idealpd.html):
In this equation, PA and PB are the partial vapour pressures of the components A and B. In any mixture of gases, each gas exerts its own pressure. This is called its partial pressure and is independent of the other gases present. Even if you took all the other gases away, the remaining gas would still be exerting its own partial pressure.
The total vapor pressure of the mixture is equal to the sum of the individual partial pressures.
The Po values are the vapour pressures of A and B if they were on their own as pure liquids.
xA and xB are the mole fractions of A and B. That is exactly what it says it is - the fraction of the total number of moles present which is A or B.
mole fraction using, for example:
The reversible process is the process that the system takes place once and returns to its original state without any change in the system or surrounding [Gordon J. Van Wylen].
The irreversible process, this process done when the system undergoes certain process it transferred from state and can not return to its original state without any change in the system or surrounding [Gordon J. Van Wylen].
Adiabatic process, this done when the system transferred from one state to another without heat transfer to surrounding [Gordon J. Van Wylen].
Isobar process, it is a process with constant pressure [Gordon J. Van Wylen].
Isothermal process, the system transferred from state to another at constant temperature [Gordon J. Van Wylen].
Isochoric process, process with constant volume [Gordon J. Van Wylen].
Figure (12 ): The thermodynamics processes [Gordon J. Van Wylen].
20. Heat of Vaporization:
Heat of vaporization or latent heat of vaporization is the amount of heat needed to transfer certain amount of matter from liquid state to vapor state. Heat of vaporization depends on the matter itself, its amount (mass), and the temperature. Table below shows the heat of vaporization of water at different temperatures [Gordon J. Van Wylen].
Heat of Vaporization
Table (1): Heat of Vaporization for water at different temperatures [Gordon J. Van Wylen].
21. Throttling Processes:
Throttling process done when fluid passing through valve or sudden reducing in area, the flow is steady and the pressure id drooped across the valve; in the throttling process the enthalpy is constant, so the throttling process is a process with constant enthalpy.
One application of throttling process is the throttling calorimeter, throttling calorimeter is a device used to determine the quality of a two phase liquid-vapor mixture [Gordon J. Van Wylen].
Figure (13 ): Throttling process [Gordon J. Van Wylen].
22. Joule Thomson Coefficient:
Joule-Thomson coefficient relates to the throttling process, it's the result of deviation of temperature drop to pressure drop for a steady state, steady flow through partially opening valve. The equation below shows Joule-Thomson coefficient:
Positive Joule-Thomson coefficient means that there is temperature drop during the throttling process, but when it is negative the temperature rises during the throttling process [Gordon J. Van Wylen].
23. Maxwell's Relations:
Maxwell relations are mathematical relations for compressible fluids, this relation are related four properties, the thermodynamics properties in Maxwell relations are pressure (P), Temperature (T), specific volume (v), and entropy (S). Maxwell relations are summarized in three views as shown below, the first view the basic equation, the second view the Maxwell relation, and the last view is the working equation [Gordon J. Van Wylen].
U: internal energy. CP: specific heat under constant pressure.
T: Temperature . Cv: specfic heat under constant specific volume.
24. Chemical equilibrium in gases:
Thermodynamics equilibrium are established when no change in macroscopic property is obtained that is mean the system is isolated from the surroundings. The equilibrium is classified to three types' mechanical equilibrium, chemical equilibrium, and thermal equilibrium. In chemical equilibrium there is no reaction or matter transfer from one part of the system to another part (P.K. NAG).
The system may be in mechanical equilibrium yet the system may undergo spontaneous Change of internal structure due to chemical potential, such as chemical reaction or a transfer of matter, the system then is said to be in chemical equilibrium if all interactions or changes in the system cease to take place. A combustion mixture of oxygen and gasoline is not in chemical equilibrium once the mixture is ignited.
25. Statements of the Second Law/ Kelvin /Planck/Clausius Statement:
It is impossible to construct a device which, operating in a cycle, will produce no effect other that raising of a weight and cooling of heat reservoir (M.L. Mathur).
It is impossible to construct a cyclic device whose effect is to extract heat from a heat reservoir and completely convert into work (M.L. Mathur).
It is impossible to construct a cyclic device which will produce no effect other than the transfer of heat from a low temperature source to high temperature heat source (M.L. Mathur).
The heat can not flow by itself (with out the help of an external agency) from low temperature to high temperature (M.L. Mathur).
Figure (14): This is not possible (Kelvin-Planck).
26. Entropy of a Mixture of Ideal Gases/ Gibbs-Dalton's Law:
The Gibbs-Dalton equation deals with gas mixture properties, the total thermodynamic property of a mixture of ideal gases is the sum of the properties that the individual gases would have if each occupied the total mixture volume alone at the mixture temperature, (M.L. Mathur), also the mathematical form of Gibbs-Dalton equation as shown below:
Specific heat under constant pressure
Specific heat under constant specific volume
Table (2): Gas mixture equations (M.L. Mathur).
Availability is the system maximum available energy. This not only depends on the given state of the system but also on the final state to which the system has to be taken and manner in which it is done. When availability of the system is required to be determined then the final state of system ought to be dead state (M.L. Mathur).
The following points should be observed when determining the availability of any system:
The final state of the system is dead state.
The system undergoes change of state by a reversible process.
The concept of availability introduce entirely a new and beneficial concept in the field of heat engines where overall thermal efficiency, obtained on the basis of total chemical energy of the fuel was the only basis for comparing engines and their performance.
28. Real Gases /Virial Equation of State /Van der Waals Equation of State:
The continuity of liquids and gases were studied by Van der Waals, the equation of equation of state for gas was obtained in 1873, and the general form of Van der Waals equation is:
a: constant measures the cohesive forces.
b: constant accounts the volume of gas molecules.
v: specific volume.
: Universal gas constant.
T: Absolute gas temperature.
The limitations of Van der Waals equation are (M.L. Mathur):
The constants a and b are measured constant for a substance where as they are not; this has been proved theoretically as well as experimentally.
The p-v plot of Van der Waals equation differs from Andrews plot.
The value of the critical volume obtained from Van der Waals equation al coefficient is 3b as compared to its experimental determine value of 2b for the moist substances.
The critical coefficient is 0.375 for Van der Waals gas equation but from experiments it was from 0.2 to 0.3 for most substances.
Fugacity (f) was used in the first time by Lewis, the value of fugacity approaches the value of pressure as the letter tends to zero, when the ideal gas conditions applies. The differential of the Gibbs function of an ideal gas undergoing an isothermal process is (P.K.NAG):
For an ideal gas the fugacity f equal the gas pressure p, fugacity has the same dimensions as pressure.
Figure (15): Fugacity with temperature.
30. Dalton's Law, Raoult's Law, Henry's Law:
Dalton states that the pressure of a mixture of gases is equal to the sum of the partial pressure of each constituent. This can be easily done using perfect gas equation for constituent as well as for the mixture (M.L. Mathur).
Raoult's law [for F. M. Raoult , a French physicist and chemist] states that the addition of solute to a liquid lessens the tendency for the liquid to become a solid or a gas, i.e., reduces the freezing point and the vapor pressure (see solution ). For example, the addition of salt to water causes the water to freeze below its normal freezing point (0Â°C) and to boil above its normal boiling point (100Â°C). Qualitatively, depression of the freezing point and reduction of the vapor pressure are due to a lowering of the concentration of water molecules, since the more solute is added, the less the percentage of water molecules in the solution as a whole and therefore the less their tendency to form into a crystal solid or to escape as a gas. Quantitatively, Raoult's law states that the solvent's vapor pressure in solution is equal to its mole fraction times its vapor pressure as a pure liquid, from which it follows that the freezing point depression and boiling point elevation are directly proportional to the modality of the solute, although the constants of proportion are different in each case. This mathematical relation, however, is accurate only for dilute solutions. The fact that an appropriate solute can both lower the freezing point and raise the boiling point of a pure liquid is the basis for year-round antifreeze for automobile cooling systems. In the winter the antifreeze lowers the freezing point of the water, preventing it from freezing at its normal freezing point; in the summer it guards against boil over by raising the boiling point of the water.
In chemistry, Henry's law is one of the gas laws, formulated by William Henry in 1803. It states that: At a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid. An equivalent way of stating the law is that the solubility of a gas in a liquid at a particular temperature is proportional to the pressure of that gas above the liquid. Henry's law has since been shown to apply for a wide range of dilute solutions, not merely those of gases. An everyday example of Henry's law is given by carbonated soft drinks. Before the bottle or can is opened, the gas above the drink is almost pure carbon dioxide at a pressure slightly higher than atmospheric pressure. The drink itself contains dissolved carbon dioxide. When the bottle or can is opened, some of this gas escapes, giving the characteristic hiss (or "pop" in the case of a champagne bottle). Because the pressure above the liquid is now lower, some of the dissolved carbon dioxide comes out of solution as bubbles. If a glass of the drink is left in the open, the concentration of carbon dioxide in solution will come into equilibrium with the carbon dioxide in the air, and the drink will go "flat" (http://en.wikipedia.org/wiki/Henry's_law).
31. Lost Work Rate, Irreversibility Rate, Availability Loss:
Entropy is produced as a result of irreversibilities present in the process, this may explained with the help of concept of lost work. The lost in work is zero in a reversible process and it increases with the increase in irreversibility of the process till it becomes maximum in case of totally irreversible process. The lost work is thus defined as the difference of work obtained in a reversible process and actual process (M.L. Mathur).
The following notes for the work lost should be taken:
For a reversible process when the work lost is zero the change in entropy is given by:
The entropy of a system can be increased by two ways, firstly by adding heat to the system or by having it undergoes an irreversible process.
The increase in entropy due to work lost is called entropy production.
For an adiabatic process, the change in entropy is associated with irreversibilities only.
32. Irreversibility and Entropy of an Isolated System:
The entropy of an isolated system can never decrease. This is known as the principle of increase of entropy. An isolated system can always be formed by including any system and its surroundings within a single boundary. Some times the original system which is then only a part of the isolated system called a subsystem. The system and surroundings together include every thing which is affected by the process (P.K. NAG).
Entropy may be decreased locally at some region within the isolated system. But it must be compensated by a greater increase of entropy some where within the system so that the net effect of an irreversible process is an entropy increase of the hole system. The entropy increase of an isolated system is a measure of the extent of an irreversibility of the process undergone by the system.
The entropy of an isolated system always increases and becomes a maximum at the state of equilibrium. When the system is at equilibrium any conceivable change entropy would be zero.
33. Reversible and Irreversible Processes:
A reversible process (ideal process) is one which is performed in such a way that at the conclusion of the process, both the system and surroundings may be restored to their initial states, with out producing any changes in the rest of the universe. Let the stare of a system be represented by A and let the system be taken to state B by following the path AB. If the system and also surroundings are restored to their initial states and no change in the universe is produced, then the process AèB will be reversible process. In the reverse process the system has to be taken from state B to A by following the same path BèA (P.K. NAG).
Any irreversible (natural) process carried out with a finite gradient is an irreversible process. A reversible process, which consists of a succession of equilibrium states, is an idealized hypothetical process.
Figure (16): Reversible process (P.K. NAG).
33. Kinetics: Reaction rates, half lives:
Half-life is the period of time it takes for a substance undergoing decay to decrease by half. The name originally was used to describe a characteristic of unstable atoms (radioactive decay), but may apply to any quantity which follows a set-rate decay.
The original term, dating to 1907, was "half-life period", which was later shortened to "half-life" sometime in the early 1950s.
Half-lives are very often used to describe quantities undergoing exponential decay-for example radioactive decay-where the half-life is constant over the whole life of the decay, and is a characteristic unit (a natural unit of scale) for the exponential decay equation. However, a half-life can also be defined for non-exponential decay processes, although in these cases the half-life varies throughout the decay process. For a general introduction and description of exponential decay, see the article exponential decay. For a general introduction and description of non-exponential decay, see the article rate law.
An exponential decay process can be described by any of the following three equivalent formulae:
N0: is the initial quantity of the thing that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
Nt : is the quantity that still remains and has not yet decayed after a time t,
t1 / 2 : is the half-life of the decaying quantity,
Ï„ : is a positive number called the mean lifetime of the decaying quantity,
Î» : is a positive number called the decay constant of the decaying quantity.
34. Temperature, pressure and DG:
Gibbs equation shows the relation between pressure, temperature, and the change in free energy. The equation below shows that relation.
G (p,T) = U + pV âˆ’ TS
which is the same as:
G (p,T) = H âˆ’ TS
U :is the internal energy (SI unit: joule)
p :is pressure (SI unit: pascal)
V :is volume (SI unit: m3)
T :is the temperature (SI unit: kelvin)
S :is the entropy (SI unit: joule per kelvin)
H :is the enthalpy (SI unit: joule)
35. Entropy and Disorder:
Work is a macroscopic concept. Work involves orderly motion of molecules as in the expansion or compression of a gas. The kinetic energy and potential energy of a system represent orderly forms of energy. The kinetic energy of a gas is due to the coordinated motion of all the molecules with the same average velocity in the same direction. The potential energy is due to vantage position taken by the molecules or displacements of molecules from their normal position.
It may state roughly that the entropy of a system is a measure of the degree of molecular disorder existing in the system.
Figure (17): Entropy and disorder (www.physcis.com).
36. Osmotic pressure / Arrhenius Law:
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the rate constant, and therefore, rate of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation for it. Nowadays it is best seen as an empirical relationship. It can be used to model the temperature-variance of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions.
A historically useful generalization supported by the Arrhenius equation is that, for many common chemical reactions at room temperature, the reaction rate doubles for every 10 degree Celsius increase in temperature (http://en.wikipedia.org/wiki/Arrhenius_equation).
In short, the Arrhenius equation gives "the dependence of the rate constant k of chemical reactions on the temperature T (in absolute temperature, such as kelvins or degrees Rankine) and activation energy Ea , as shown below:
37. Partition functions:
In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives.
There are actually several different types of partition functions, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy.) The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances (http://en.wikipedia.org).
pi: indicate particle momenta.
xi :indicate particle positions.
d3 :is a shorthand notation serving as a reminder that the pi and xi are vectors in three dimensional space.
38. Le Chatelier's principle for Temperature:
In 1884, the French Chemist Henri Le Chatelier suggested that equilibrium systems tend to compensate for the effects of perturbing influences. When a system at equilibrium is disturbed, the equilibrium position will shift in the direction which tends to minimize, or counteract, the effect of the disturbance (http://en.wikipedia.org).
If the concentration of a solute reactant is increased, the equilibrium position shifts to use up the added reactants by producing more products.
If the pressure on an equilibrium system is increased, then the equilibrium position shifts to reduce the pressure.
If the volume of a gaseous equilibrium system is reduced (equivalent to an increase in pressure) then the equilibrium position shifts to increase the volume (equivalent to a decrease in pressure)
If the temperature of an endothermic equilibrium system is increased, the equilibrium position shifts to use up the heat by producing more products.
If the temperature of an exothermic equilibrium system is increased, the equilibrium position shifts to use up the heat by producing more reactants.
39. Colligative properties:
Colligative properties are the properties of the solution based on the number of molecules per unit volume of the solution. Colligative properties include the vapor pressure, boiling and freezing point, and osmotic pressure (http://en.wikipedia.org).
The vapor pressure of an ideal solution is dependent on the vapor pressure of each chemical component and the mole fraction of the component present in the solution.
The boiling temperature of the solution before reaching the vapor phase, the freezing point is the lowest temperature of the solution before it transferred to solid state.
The osmotic pressure of a dilute solution at constant temperature is directly proportional to its concentration. The osmotic pressure of a solution is directly proportional to its absolute temperature.
40. Entropy and the Clausius inequality:
The second law of thermodynamics leads to the definition of a new property called entropy, a quantitative measure of microscopic disorder for a system. Entropy is a measure of energy that is no longer available to perform useful work within the current environment. To obtain the working definition of entropy and, thus, the second law, let's derive the Clausius inequality. Consider a heat reservoir giving up heat to a reversible heat engine, which in turn gives up heat to a piston-cylinder device as shown below (http://en.wikipedia.org).