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MUHAMMAD NASIR CHEMICAL ENGINEERING
Property relation for homogeneous phaseAccording of first law for a close system for n moles ------1 We also know that
together these three equation
U,S and V are molar valves of internal energy ,entroply and valume.Combing effect of both laws 1ST and 2nd Derived for reversible reactionBut contain the property of the state of the system not process of the systemConstant mass that result of differenital change from one equiriblrium state to another.Nature of the system cannot be relaxed
Conti.We know that H = U +PV GIBBS energy and Helmholtz energy equation A = U- TS----2 G= H-TS ----3 Putting the values on above equation so this equation becomes
Conti. When d(nU) replace by equation no 1 equation becomes 4
Same way multiplying equation 2 and 3 by n and takind differtional equations becomes 5 and 6
Conti.Equation 5 and 6 are subject of resection of equation 1.for the case of one mole of homogeneous fluid at constant pressureThese are fundamental equations for homogeneous equaions
Another set of equations follow from equation 6 and 7 for exactness for a differtional expression for a function f(x.y)
As result we get new sets of equations
Entropy:Is a measure of disorder or randomness of a system. An ordered system has lowentropy. A disordered system hashighentropy.
Enthalpy:Is defined as the sum of internal energy of a system and the product of the pressure and volume of the system. The change in enthalpy is the sum of the change in the internal energy and the work done.
Enthalpy and entropy are different quantities. Enthalpy has the units of heat, joules. Entropy has the units of heat divided by temperature, joules per kelvin
Enthalpy Vs. Entropy
Enthalpy:It is donated by 'H', refers to the measure of total heat content in a thermodynamic system under constant pressure. Enthalpy is calculated H = E + PV(where E is the internal energy). The SI unit of enthalpy is joules (J).
Entropy:It is denoted by 'S', refers to the measure of the level of disorder in a thermodynamic system. Entropy is calculated S = Q/T (where Q is the heat content and T is the temperature).It is measured as joules per kelvin (J/K).
Relationship between Enthalpy and Entropy of a Closed System(T.S=H)
Here, T is the absolute temperature, H is the change in enthalpy, and S is the change in entropy. According to this equation, an increase in the enthalpy of a system causes an increase in its entropy.
How does entropy change with pressure?Theentropyof a system decreases with an increase in pressure.Entropy is a measure of how much the energy of atoms and molecules become more spread out in a process.
If we increase the pressure on the system, the volume decreases. The energies of the particles are in a smaller space, so they are less spread out. The entropy decreases.
If we decrease the pressure on the system, the volume increases. The energies of the particles are in a bigger space, so they are more spread out. The entropy increases.
Pressure Dependence of EntropyFor solids and liquids entropy change with respect to pressure is negligible on an isothermal path. This is because the work done by the surroundings on liquids and solids is miniscule owing to very small change in volume. For ideal gas we can readily calculate the entropy dependence on the pressure as follows
Temperature dependence of Entropy
Using the usual conditions such as isobaric or isochoric paths we can see that:
Just as in case of H the above formulae apply as long as system remains in single phase. On the other hand if system undergoes a phase transition, at constant temperature and pressure.
Enthalpy & Entropy as function of Temp & pressureThe most useful property relation for the Enthalpy and Entropy of a homogenous phase result when these properties expressed as function of T & P
We need to know how H & S vary with Temperature and Pressure.
Consider First the Temperature derivative. Equation 2.2 divide the heat capacity at constant pressure.
Another Expression for this quantity is obtained by division of Eq. (6.8) by dT and restriction of the result to Constant P.
Combination of this equation with Eq (2.2) gives
The pressure derivative of the entropy results directly from Eq. (6.16)
The Corresponding derivative for the enthalpy is found by division of Eq. (6.8) by dP and restriction to constant T.
As a Result of Equation (6.18) this become
The functional relation chosen here for H & S are H = H(T , P)S = S(T , P)
The partial derivative are given by Eqs. (2.20) and (6.17) through (6.19)
These are general Equation relating the Enthalpy and Entropy of homogenous fluid at constant composition to Temperature and pressure.
Internal energy as a function of P
Internal energy is given as U = H PV Differentiation yields
As we know
Now by putting this equation in above equation
The ideal gas stateAs we know ideal gas
By differentiating with respect to T and keeping P constant
Now substituting this equation into following equations
We got following equations
Alternative forms for Liquids
in following equations
We got following
whenIn following equation
Internal energy and entropy as a function of T & P
It is applied to a state a constant volumeAlternate form
Gibbs free energy
Theenergy associated with a chemical reaction that can be used to do work. The freeenergyof a system is the sum of its enthalpy (H) plus the product of the temperature (Kelvin) and the entropy (S) of the system:
According to thesecond law of thermodynamics, for systems reacting atSTP(or any other fixed temperature and pressure), there is a general natural tendency to achieve a minimum of the Gibbs free energy
The Gibbs free energy is:
which is the same as:
where:Uis theinternal energy(SI unit:joule)pispressure(SI unit:pascal)Visvolume(SI unit: m3)Tis thetemperature(SI unit:kelvin)Sis theentropy(SI unit: joule per kelvin)His theenthalpy(SI unit: joule)
The Gibbs free energytotal differentialnatural variablesmay be derived viaLegendre transformsof theinternal energy.
Where, iis thechemical potentialof theithchemical component. (SI unit: joules per particleor joules per mole Niis thenumber of particles(or number of moles) composing theith chemical component
The definition ofGfrom above is
Taking the total differential, we have
Replacing dUwith the result from the first law gives
Applications of Gibbs Free EnergyColligative properties of solutionsBoiling point elevation and freezing point depressionThe pressure on a liquid affects its volatilityElectron-free energy levels
Effect of pressure on a liquid Applying hydrostatic pressure to a liquid increases the spacing of its microstates, so that the number of energetically accessible states in the gas, al though unchanged, is relatively greater thus increasing the tendency of molecules to escape into the vapor phase. In terms of free energy, the higher pressure raises the free energy of the liquid, but does not affect that of the gas phase.
Thermodynamics of rubber bands
Rubber is composed of random-length chains of polymerizedisoprene molecules. The poly(isoprene) chains are held together partly by weak intermolecular forces, but are joined at irregular intervals by covalent disulfide bonds so as to form a network..
ContiThe intermolecular forces between the chain fragments tend to curl them up, but application of a tensile force can cause them to elongate The disulfide cross-links prevent the chains from slipping apart from one another, thus maintaining the physical integrity of the material. Without this cross-linking, the polymer chains would behave more like a pile of spaghetti.
ExampleHold a rubber band (the thicker the better) against your upper lip, and notice how the temperature changes when the band is stretched, and then again when it is allowed to contract.a)Use the results of this observation to determine the signs of H, Gand Sfor the process rubberstretched rubberunstretchedb)How will the tendency of the stretched rubber to contract be changed if the temperature is raised?
Solution a) Contraction is obviously a spontaneous process, so Gis negative.You will have observed that heat is given off when the band is stretched, meaning that contraction is endothermic, so H> 0. Thus according to G= HTS,Sfor the contraction process must be positive.b)Because S>0, contraction of the rubber becomes more spontaneous as the temperature is raised.