Chapter 4 thermodynamic functions and fundamental equations

Internal energy U: minimum at a given S, V. Entropy S: maximum at a given U, V. Gibbs function G: minimum at a given T, P. Helmholtz function F: minimum at a given T, V. Enthalpy H: minimum at a given S, P.

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Ngo Thanh An

PHYSICAL CHEMISTRY 1

Chapter 4 – Thermodynamic functions and

Fundamental equations

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Part 1 – Thermodynamic functions

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Thermodynamic function Symbol Definition unit

Gibbs free energy (isothermal, isobaric

thermodynamic potential) G G = H –TS Cal or J

Helmholtz free energy (isothermal,

isochoric thermodynamic potential) F F = U – TS Cal or J Entropy S dS=Qrev/T cal.KJ.K–1–1Internal energy U Cal or J Enthalpy H H = U + PV Cal or J

Thermodynamic functions

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Thermodynamic functions

V

U S

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• Internal energy U: minimum at a given S, V.

• Entropy S: maximum at a given U, V

• Gibbs function G: minimum at a given T, P

• Helmholtz function F: minimum at a given T, V

• Enthalpy H: minimum at a given S, P

Characteristics of thermodynamic functions

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Theo nguyên lý tăng entropy, khi entropy đạt cực đại thì:

Ta đặt giá trị A bằng: Áp dụng công thức:

Vậy A sẽ bằng:

Ta lại có:

Chứng minh

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Vậy giá trị A cũng sẽ bằng 0, tức là U cũng sẽ đạt cực trị theo V tại 1 giá trị entropy nào đó Ta xem A là một hàm số của A = A(V, U(V)) Tính chất đạo hàm của hàm hợp cho ta công thức:

Như vậy, ta sẽ có:

Với điều kiện A = 0, sẽ cho ta:

Chứng minh

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Như vậy, hàm U sẽ đạt cực tiểu

Chứng minh

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If we have a function F = F(x,y)

We need to transform function F(x,y) into:

 Function G(x,w) where w is a conjugate variable of variable y

 Function H(u,y) where u is a conjugate variable of variable x

 Function L(u,w) where u, w are conjugate variables of x, y respectively

(1)

where:

(2)Equation (1) – equation (2), having :

Where

Prove:

Relations of thermodynamic functions

Legendre transform

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where:

In summary, Legendre transform is a method to converse a function F(x,y) into a

new function G(x,w), where y and w are a couple of conjugate variable

Legendre transform

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From function U = U(S, V), we

can transform to different state

Application of Legendre transform

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Relationship between state function and

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Josiah Willard Gibbs

Hermann von Helmholtz

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•

Applications of thermodynamic functions

First order differential

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Second order differential

Applications of thermodynamic functions

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c Trongtrườnghợpápdụngphươngtrình Gibbs chohàm F:

• Ta sẽcó:

d Trongtrườnghợpápdụngphươngtrình Gibbs chohàm G:

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Effect of thermodynamic properties

Summary

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Internal Energy Changes

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Enthalpy Changes

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25

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Entropy Changes

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The temperature of a fluid may increase,

decrease, or remain constant during a

throttling process The development of an h = constant line on a P-T diagram.

The temperature behavior of a fluid during a throttling (h = constant) process is

described by the Joule-Thomson coefficient

The Joule-Thomson coefficient

represents the slope of h = constant

lines on a T-P diagram.

Joule – Thomson coefficient

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Constant-enthalpy lines of a substance

However, the fluid temperature decreases during a throttling process that takes place on the left-hand side

of the inversion line

It is clear from this diagram that a cooling effect cannot be achieved by throttling unless the fluid is below its maximum inversion temperature

This presents a problem for substances whose maximum inversion temperature is well below room temperature

Joule – Thomson coefficient

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Part 2 – Fundamental equations

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The equation is obtained by combining the first and second law of thermodynamics

a) First law of thermodynamics: dU = Q - A

Second law of thermodynamics

dU  T.dS - A

Q T

dS � 

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U H F G

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