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Thermodynamics; Introduction

Thermodynamics is used to describe heat and energy changes and chemical equilibria. It allows the prediction of equilibria positions, but NOT the rates of reaction or rates of change.


V The temperature scale and absolute zero

V The Zeroth Law of Thermodynamics

V The First Law of Thermodynamics
V The Second Law of Thermodynamics
V The Third Law of Thermodynamics

V Chemical potential
V Internal energy

V Heat capacity and specific heat

V Helmholtz free energy


'classical thermodynamics ...... is the only physical theory of universal                                                         content which I am convinced will never be overthrown'

Albert Einstein 


This is an elementary introduction to the terms and ideas of thermodynamics. Of great importance is that the laws of thermodynamics are absolute laws of the Universe and cannot be circumvented. Remember this, and you will not be misled by purveyors of 'too good to be true' ideas such as running cars on just water. Thermodynamics describes the energy content of systems at equilibrium and their transformations and is independent of the pathway chosen. Thermodynamics is not to be confused with kinetics, which describes the rate of change of processes and their mechanisms and depends on the pathway chosen.


Heat is the amount of energy flowing from one body of matter to another spontaneously due to their temperature difference or by any means other than through work or the transfer of matter.


Temperature is a quantitative measure of hot and cold (K). Hotness is a body's ability to impart energy as heat to another body that is colder. Several temperature scales have been described, but the SI scale [1031] uses the kelvin.


Pressure is the force per unit area (Pa). It may be positive or negative and is not, in itself, directional. A positive pressure indicates that the system tends to expand whereas a negative pressure tends to contract.


Density of any substance is its mass (kg) divided by its volume (m3).


Energy is difficult to define precisely being based on the Laws of thermodynamics. In different circumstances, it can be the capacity to do work, or the capacity to provide heat or radiation. There can be a conversion between different forms of energy, but energy cannot be created or destroyed.


Work is the energy associated with the action of a force.


Symbols. The symbol Δ (capital delta) means the change between the start and end states (see below). The symbol δ (small delta) means " a small change in". The symbol d (dee) is a differential (meaning "an infinitesimal change in"). The symbol ∂ (partial dee) is a mathematical symbol to denote a partial derivative; meaning the change in a function of several variables with respect to one of those variables, under some constant conditions (usually the remaining conditions) as stated in the subscript(s). Thus, (∂H/∂T)P is the partial derivative of H with respect to T under conditions of constant P (see below).


dP=(dP/dT)v. dT+(dP/dV)T. dV


Although advanced thermodynamics can appear daunting when first encountered, there are just three primary concepts: energy, entropy, and absolute temperature.

The temperature scale and absolute zero

Temperature is a numerical scale of relative hotness versus coldness. It is an intensive physical quantity for a material (does not depend on the amount of substance present in the system) and, when it increases, it indicates that the material has increased its energy content. The temperature may usually be considered as a measure for the average kinetic energy of particles. Absolute zero is the lowest limit for the thermodynamic temperature scale; nothing can be colder. It is defined as zero kelvin (0 K) and cannot be reached. The temperature scale did define the triple point of water (a triple point is a singular state with its own unique and invariant temperature and pressure) as +273.16 K precisely, and this set the kelvin temperature scale and the size of the kelvin. However, the kelvin is now redefined in terms of the Boltzmann constant (kB, J ˣ K−1), the second, the meter and the kilogram.


     One kelvin = 1.380 649 ˣ 10−23 J ˣ kB−1      (exactly)

                        One kelvin = 1.380 649 ˣ 10−23 ˣ kg ˣ m2 ˣ s−2 ˣ kB−1      [2395].


The closest approach to absolute zero achieved so far is about +0.000 000 001 K.

Internal energy

Thermodynamics introduces a term U that is the INTERNAL ENERGY; the energy contained within the system, including its vibrational energy and bonding and interactional energy. This depends on the number of its accessible quantum states and its volume at a given pressure.


ΔU is the change in the Internal energy; ΔU equals the heat added to a system less the work done by the system.

The Zeroth Law of Thermodynamics

The Zeroth Law of Thermodynamics states that two bodies in contact will come to the same temperature. It follows that If a body A is in thermal equilibrium with two other bodies, B, and C, then B and C are in thermal equilibrium with one another.

The First Law of Thermodynamics

The First Law of Thermodynamics is the law of conservation of energy:

                                            'Energy can neither be created nor destroyed'


It can be expressed in everyday terms:


                                           You can't win, you can only break even

                                           You do not get anything for nothing
                                           There is no such thing as a free lunch

                                           The energy of the Universe is constant


It states that the energy in an isolated closed system is conserved, where energy is the capacity to do work. Heat energy can do work by (for example) changing a temperature or pressure. The isolated system may be a chemical reaction, a natural process, a cell, the earth, etc., If these systems are isolated, neither energy nor matter can enter or leave.


For an isolated closed system, any change in the internal energy ΔU is composed of the exchanged heat ΔQ and work ΔW done on or by the system,

ΔU = ΔQ + ΔW


and for a reversible process, the heat energy change ΔQ = TΔS and the work done ΔW = -PΔV ; (for non-spontaneous reactions , see the discussion at [3735]).





Thermodynamics introduces a term H that is the ENTHALPY; a measure of the heat content of the system (with units J ˣ mol−1).

    ΔH is the CHANGE IN ENTHALPY; the heat lost or gained


H = U + PV


In many biological systems, H = U as pressures and volumes, and their changes, are small.


Under constant-pressure conditions, the change in enthalpy is given



By convention:

    ΔH is negative when heat is released by the system; such as in exothermic processes

    ΔH is positive when heat is absorbed by the system; such as in endothermic processes


In a sequence of reactions the overall change in enthalpy is the sum of the enthalpies involved:

ΔHoverall = Σ ΔH

             A = B             ΔH1                     e.g.         C + ½O2       ->   CO            ΔH1 
             B = C             ΔH2                     e.g.         CO + ½O2    ->   CO2           ΔH2                 
sum      A+B = B+C    ΔH1 + ΔH2                            C + O2        ->   CO2        total ΔH = ΔH1 + ΔH2    


However, ΔH does not tell us if or how fast the process will go: e.g.,

            desk burning;       wood + O2     ->     CO2 + H2O        ΔH is negative, and heat is given out

We know that a desk will not spontaneously burn as the reaction is incredibly slow. It would burn if we created a fire'


            melting of ice;                   ice      ->    water                 ΔH is positive, and heat is absorbed

We know that ice will melt if the temperature is above 0 °C.


Therefore an enthalpy change, by itself, cannot predict the direction of a process.

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The Second Law of Thermodynamics

The Second Law of Thermodynamics tells us about the direction of processes; a hot coffee gets colder if it is let to stand—it never gets hotter. The Second Law of Thermodynamics states that the total disorder in an isolated system can only increase over time.


It can be expressed in everyday terms:


                                           You can't even break even (except at absolute zero)
                                           The house always wins
                                           The Universe is becoming more chaotic

                                           Disorder within the Universe always increases with time
                                           A perpetual motion machine cannot be built

                                           No process for converting heat into energy is 100 % efficient

                                           Heat only spontaneously flows from a hot object to a cold one, not from cold to hot




Entropy explanation



It states that the total order in an isolated system cannot increase over time. If part of such a system becomes more ordered, other parts must become even more disordered. In ideal cases, the amount of order may remain constant. There is only one way in which the entropy of a supposedly closed system can be decreased, and that is to transfer heat from the system (that is not closed).


Thermodynamics introduces a term S that is the ENTROPY; a measure of disorder and chaos of the system (a measure of the number of microscopic states of a system with units of J ˣ mol−1 ˣ K−1). In a simple equi-probable system, it may be defined as


S0 = kB ˣ Ln(N)


where kB is the Boltzmann constant, LN() is the natural logarithm, and N is the number of configurations.

    ΔS is the CHANGE IN ENTROPY; the change in order or disorder


ΔS = Σ{SProduct } − Σ{SReagent}

ΔSoverall = Σ ΔS

By convention:  

    ΔS is positive when the disorder increases; the system is more chaotic and disorganized; e.g., a liquid turning into a gas.

    ΔS is negative when order increases; the system is more ordered and organized; e.g., a liquid turning into a crystalline solid.


If there is no change in enthalpy but a process proceeds, there must be an increase in entropy; e.g., gases mixing.

If two systems are combined, the final entropy is greater than the sum of the parts.


Entropy change, by itself, cannot predict the direction of a process


                             2H2 + O2    ->      2H2O       clearly goes with negative entropy change as
     3 molecules of a mixture     ->      2 molecule of the same product


Thus, this is a process that proceeds to give a more ordered product. However, a large amount of heat is produced (negative enthalpy change) that increases the kinetic energy and disorder in the products and the surroundings.


Therefore an entropy change, by itself, cannot predict the direction of a process.

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The Third Law of Thermodynamics

The Third Law of Thermodynamics addresses the problem concerning the direction of a process. It follows from a combination of the First and Second Laws. The entropy of a perfect crystal at absolute zero is precisely equal to zero.


It can be expressed in everyday terms:


                                          You can't reach absolute zero

                                          You can't stay out of the game
                                          The First and Second Laws cannot be got around


Thermodynamics introduces a term G is the GIBBS FREE ENERGY c (available energy, with units J ˣ mol−1); the ability to do work of the system at constant temperature and pressure. G is sometimes called, just, the 'free energy' or 'Gibbs energy'. b


G = U + PV - TS

G = H - TS

    ΔG is the change in the Gibbs free energy. Gibbs free energy can do work at constant temperature and pressure. It determines the direction of a conceivable chemical or physical process, and is zero when a system is at equilibrium at constant temperature and pressure.


In living systems (constant temperature and pressure);


ΔG = ΔH - T ΔS


    ΔG is the maximum work obtainable from a process
    ΔG is negative when the system is able to proceed; the process is exergonic, and

        there is a positive flow of energy from the system to the surroundings
    ΔG is positive when the system is unable to proceed; the process is endergonic, and
       it takes more energy to start the reaction than what you get out of it.

    ΔG is zero when the system is at equilibrium


Every reaction has a characteristic ΔG under defined conditions. Under standard conditions (usually 1 M reactants and products, 298.15 K (25 °C), 100 kPa), this is called the Standard Free Energy change and given the symbol, ΔG°.

Where the pH = 7, rather than [H+] = 1 M this is given the symbol, ΔG°'

For the reaction  A + B = C + D

Free energy equation

where R = gas constant (8.31 J ˣ mol−1 ˣ K−1), T is the temperature (in kelvin), and Ln is the natural logarithm;


Given ΔG is zero when the system is in equilibrium, therefore

ΔG°'= - RTLn (Keq°')


In a sequence of reactions:

ΔGoverall = Σ ΔG


                        A + B = C + D           ΔG1
                        C + E = F                  ΔG2
sum          A + B + E = D + F            ΔG = ΔG1 + ΔG2


So long as the overall ΔG is negative the reaction will go from left to right; A + B + E -> D + F

As an example;

                                                                                                                    ΔG°', kJ ˣ mol−1
                          Glucose + phosphate = Glucose-6-phosphate + H2O             +13.8
                                         ATP + H2O = ADP + phosphate                              - 30.5
                                    Glucose + ATP = ADP + Glucose-6-phosphate             - 16.7


Under standard conditions (at pH 7), the process directions are determined by ΔG°';

Glucose-6-phosphate + H2O -> Glucose + phosphate

ATP + H2O -> ADP + phosphate  

Glucose + ATP -> ADP + Glucose-6-phosphate  


The ATP hydrolysis pulls the phosphorylation of the glucose.


ΔG depends on the concentration of the reactants and products as well as the temperature and ΔG°'.


Process direction depends on ΔG


The fundamental equation of thermodynamics is


ΔG = V ˣ ΔP - S ˣ ΔT + ΔnA ˣ μA + ΔnB ˣ μB + ΔnC ˣ μC +. . . . . . . .

ΔG = V ˣ ΔP - S ˣ ΔT + Σi Δni ˣ μi                                                        


At a surface, a further term must be added to the right-hand-side; +γΔA where γ (J ˣ m−2) and A (m2) are the surface tension and surface area respectively.


There are four Maxwell relations involving the second derivatives of each of the four thermodynamic potentials, U, H, F, and G.

partial dee T/partial dee V
partial dee T/partial dee P
partial dee S/partial dee V
partial dee S/partial dee P


partials relationship

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The chemical potential of substance A

The chemical potential of substance A
Chemical potential

The chemical potential (μ) is a term first used by Willard Gibbs and is the same as the molar Gibbs free energy of formation, ΔGf, for a pure substance, with units of energy (J ˣ mol−1). At equilibrium, the chemical
potential μi of individual components (i) is the same in all phases


For materials in a mixture, the chemical potential (μ) is the partial molar Gibbs free energy

chemical potential is thepartial molar Gibbs energy

where the substance changing is A with nA representing the number of A molecules, and nB represents the number of molecules of all other materials present. It follows that,


dmui/dP=Vi       and         dmui/dT=Si

and                                                             dμ = -Sm dT + Vm dP                                           (The Gibbs equation)


where Sm and Vm are the molar entropy and volume of the substance.


The total Gibbs free energy of a mixture of A and B is


G = nA ˣ μA + nB ˣ μB


The chemical potential (μ) is related to its activity (a) by


μ = μ0 + RT ln(a)


where μ0 is a constant (the standard state), given values for T and P. The activity is a measure of the "effective concentration" of a material in a mixture. It is without units as it is always divided by the respective standard activity in the same units (e.g., a single concentration unit; often = 1 mol ˣ L−1). a


The chemical potential of an aqueous solution is given by


μ = μ0 + RT ln(aw)

μ = μ0 + RT ln(xw)


where xw is the mole fraction of the ‘free’ water, aw is the water activity, and μ0 is the chemical potential of the pure water. For a graph of μ versus aw see elsewhere.


When two or more phases are in equilibrium, the chemical potential of a substance is the same in each of the two phases and at all points in each of the phases. If a material is distributed between two phases in equilibrium;

μA - μB = RT ln(aA/aB)


The molar chemical potential of water in an electrolyte solution (molality m; activity a),H2O(T, P, m), can be written as


H2O(T, P, m)= H2O(T, P, 0) + RT ln {aH2O(T, P, m)}

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Heat capacity and specific heat

The heat capacity (C) is the ratio of the measurable heat added (Q) or subtracted to an object to the resulting temperature change




with units J ˣ K−1). The specific heat (mass-specific heat capacity) is the heat capacity per unit mass of a material. CP and CV are the heat capacities at constant pressure (isobaric, CP = (∂U/∂T)P and constant volume (isochoric, CV = (∂U/∂T)V) respectively. The specific heat is the amount of heat needed to raise the temperature of one kilogram of mass by 1 kelvin. The molar heat capacity is the heat capacity per mole of a pure substance.


As                 H = U + PV

δQ = δU + PδV

CP = (∂H/∂T)P

CV = (∂U/∂T)V


where U is the internal energy and H is the system enthalpy.


The full expressions are


CP = (∂H/∂T)P= T(∂S/∂T)P = <(ΔS)2>TP /(kBN) = <(ΔH)2>TPN /(kBT2) = (∂U/∂T)P + PVαV 


 CV = (∂U/∂T)V= T(∂S/∂T)V = (∂H/∂T)V - PVαP = CP - TVαP2/βT




CP/CV = βT/βS > 1


where kB, P, T, N, V, H, S, αV, αP, βT, βS and are the Boltzmann constant, pressure (Pa), temperature (K), number of molecules, specific volume (V = 1/density; m3 mol−1), enthalpy, entropy, isobaric cubic expansion coefficient (αV=(∂V/∂T)P/V; K−1), relative pressure coefficient (αP=(∂P/∂T)V/P; K−1), isothermal compressibility (βT=-(∂V/∂P)T/V; Pa−1) and adiabatic compressibility (βS=-(∂V/∂P)S/V; Pa−1) respectively; the <> brackets indicate the fluctuations in the values about their mean values. (also see [1481]). Thus, the heat capacity is a measure of the entropy fluctuation ΔS of N molecules at constant temperature and pressure. The isothermal compressibility is a measure of the fluctuations ΔV of the mean volume V occupied by a given number of molecules. The expansion coefficient is related to the correlations between the entropy and volume fluctuations. In contrast to most other liquids, in which entropy and volume fluctuations are positively correlated, ΔS and ΔV are anti-correlated in water below 4 °C, with a decline in volume fluctuations associated with an enhancement of entropy fluctuations.

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a Note: You are only allowed to take the logarithm of a dimensionless positive number, not of a physical quantity of unit dimensions (like m, s, L, mol, kg, V, etc.,). Thus when taking logarithms, a ≡ a/a0 where a0 is standardized as one unit in the dimensions of a. In the same manner, you cannot use trigonometric nor exponential functions of physical quantities given in dimensions. [Back]


b Helmholtz free energy (A). Under conditions of where no pressure/volume work can be extracted (A ≤ G) but where pressure may change (a closed thermodynamic system, e.g., in closed explosions), G = U + PV - TS becomes.


A = U - TS


where A is the Helmholtz free energy c (often given in Physics textbooks as F) and as for reversible reactions,


dU=TdS - pdV        and        d(TS) =TdS +SdT

dA = -SdT - pdV



c It has been recommended that 'Gibbs free energy' and 'Helmholtz free energy' be known simply as 'Gibbs energy' [IUPAC] and 'Helmholtz energy' [IUPAC] but, at present, there is a scientific consensus to keep the historical terms. [Back]

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