MUHAMMAD NASIR CHEMICAL ENGINEERING

Thermodynamic -1

Property relation for homogeneous phase

According of first law for a close system for n moles

------1

We also know that

together these three equation

Conti……….

U,S and V are molar valves of internal energy ,entroply and valume.

Combing effect of both laws 1ST and 2nd

Derived for reversible reaction

But contain the property of the state of the system not process of the system

Constant 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 pressure

These 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)

Conti……….

As result we get new sets of equations

Entropy: Is a measure of disorder or randomness of a system. An ordered system has low entropy. A disordered system has high entropy.

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. EntropyEnthalpy: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 + P∆V(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.

Entropy

pathleirreversibforS

pathcyclicreversibleforTdqS

0

0

How does entropy change with pressure?

The entropy of 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

1

2ln

.

0)(

0.

PP

nRS

PdPnR

TdPVdS

dPPVdVVdPPdVPVd

ETdVP

Tdw

Tdq

dS rev

Temperature dependence of Entropy

Using the usual conditions such as isobaric or isochoric paths we can see that:

Just as in case of ΔH 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.

pathisobaricTTC

TdTC

TdTC

TdqS

pathisochroicTTC

TdTC

TdTC

TdqS

PPP

vvv

1

2

1

2

ln

ln

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.

PTH

PTS

TPH

TPS

PP

CTH

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.

PP TST

TH

TC

TS P

P

PT TV

PS

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)

VPST

PH

TT

dPPHdT

THdH

TP

dPPSdT

TSdS

TP

PT TVTV

PH

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.

dPTVTVdTCdH

PP

dPTV

TdTCdS

PP

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 state As 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

Following

when

In following equation

We obtain

Internal energy and entropy as a function of T & PAs we know dQ……………….1

dU

Putting the of aqua 1 into aqua 2

dU….3

Temperature and volume often serve as more convenient independent variables hen do temperature pressure. The most useful relation are hen for infernal energy and entropy. Required derivatives are

()v ,()t ,()v ,()t

taking he derivative of aqua 2 wish respect to temperature and volume at constant volume and temperature.

( )v =T()v………………………….4

Cv= T()v

CV/T=()v

( )t =T()t-P………………….……5

( )t =T()t-P

the chosen function here are

U=U(T,V) S=S(T,V)

taking derivative

dU= ( )v dT+ ( )t dV………….6

dS=()v dT+ ()t dV…………….7

let

dU= ( )v dT+ ( )t dV

dU= Cv dT+[T ()-P]dV

dS=()v dT+ ()t dV

dS= CV/TdT+ ()vdV

As we know

()v=

It is applied to a state a constant volume

Alternate form

Gibbs free energy

 The energy associated with a chemical reaction that can be used to do work. The free energy of 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 the second law of thermodynamics, for systems reacting at STP (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: 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)

Derivation

The Gibbs free energy total differential natural variables may be derived via Legendre transforms of the internal energy.

Where, μi is the chemical potential of the ith chemical component. (SI unit: joules per particle or joules per mole

Ni is the number of particles (or number of moles) composing the ith chemical component

     The definition of G from above is

Taking the total differential, we have

Replacing dU with the result from the first law gives

Applications of  Gibbs Free Energy

Colligative properties of solutions

Boiling point elevation and freezing point depression

The pressure on a liquid affects its volatility

Electron-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 polymerized isoprene 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..

Conti’The 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 ΔH, ΔG and ΔS for the process

rubberstretched → rubberunstretched

b) 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 ΔG is negative.You will have observed that heat is given off when the band is stretched, meaning that contraction is endothermic, so ΔH > 0. Thus according to ΔG  =  ΔH  – TΔS, ΔS for the contraction process must be positive.

b) Because ΔS > 0, contraction of the rubber becomes more spontaneous as the temperature is raised.