Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 17, Problem 17.21E
Interpretation Introduction

(a)

Interpretation:

The expression for Φ explicitly is to be stated. The given expressions are to be verified.

Concept introduction:

A mathematical function represents the way in which its value depends on some variable. A function can contain some variables and constants. The term multiplied by a variable is called its coefficient. When derivative of a function is taken with respect to a variable, then the terms that only contain other variables are considered as zero.

Expert Solution
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Answer to Problem 17.21E

The expression for Φ explicitly is represented as,

Φ=C1ξ1+C2ξ2+C3ξ3

The given expressions are verified as shown below.

Explanation of Solution

The given function ϕ can be expressed as,

Φ=i=13Ciξi …(1)

Where,

ξi represents the ith variable

Ci represents the value of the coefficient multiplying variable ξi.

The expanded form of the equation (1) is represented as,

Φ=C1ξ1+C2ξ2+C3ξ3 …(2)

Derivate equation (2) with respect to ξ1.

δΦδξ1=δδξ1(C1ξ1+C2ξ2+C3ξ3)=δδξ1(C1ξ1)+δδξ1(C2ξ2)+δδξ1(C3ξ3)

The term ξ2, and ξ3 are constant with respect to ξ1. Therefore, their derivative with respect to ξ1 will be zero.

δΦδξ1=C1+0+0=C1

Derivate equation (2) with respect to ξ2.

δΦδξ2=δδξ2(C1ξ1+C2ξ2+C3ξ3)=δδξ2(C1ξ1)+δδξ2(C2ξ2)+δδξ2(C3ξ3)

The term ξ1, and ξ3 are constant with respect to ξ2. Therefore, their derivative with respect to ξ1 will be zero.

δΦδξ2=0+C2+0=C2

Derivate equation (2) with respect to ξ3.

δΦδξ3=δδξ3(C1ξ1+C2ξ2+C3ξ3)=δδξ3(C1ξ1)+δδξ3(C2ξ2)+δδξ3(C3ξ3)

The term ξ2, and ξ1 are constants with respect to ξ3. Therefore, their derivative with respect to ξ1 will be zero.

δΦδξ3=0+0+C3=C3

Therefore, the given expressions are verified.

Conclusion

The expression for Φ explicitly is represented as,

Φ=C1ξ1+C2ξ2+C3ξ3

The given expressions are verified.

Interpretation Introduction

(b)

Interpretation:

The general expression for the derivative of Φ with respect to ξi is to be stated. The corresponding expression is to be compared with the equation 17.15. The explanation for the derivation of the equation 17.15 from the preceding equation is to be stated.

Concept introduction:

A mathematical function represents the way in which its value depends on some variable. A function can contain some variables and constants. The term multiplied by a variable is called its coefficient. When derivative of a function is taken with respect to a variable, then the terms that only contain other variables are considered as zero.

Expert Solution
Check Mark

Answer to Problem 17.21E

A general expression for the derivative of Φ with respect to ξi is represented as,

δΦδξ1=Ci

Where,

i represents a number that can be either 1,2, or 3.

The general expression for the derivative of Φ with respect to ξi and the given equation 17.15 are similar.

In the given expression it was assumed that for a large system Ni=Nj and gj is rewritten as gi. After the assumption, the equation was derived to get the equation 17.15.

Explanation of Solution

The given function ϕ can be expressed as,

Φ=i=13Ciξi …(1)

Where,

ξi represents the ith variable

Ci represents the value of the coefficient multiplying variable ξi.

A general expression for the derivative of Φ with respect to ξi is represented as,

δΦδξ1=Ci

Where,

i represents a number that can be either 1,2, or 3.

The given equation 17.15 is shown below as,

lngiNi1+αβi=0 i=1,2,3,... …(3)

Where,

αand β represents the constant.

Ni represent ith a number of the particle.

Nj represent jth a number of the particle.

gj represents degeneracy for jth the number.

gi represents degeneracy for ith the number.

The expression from which equation 17.15 is derived is shown below as,

Nj[(NlnN+jNjlngjNj)+αiNiβiNii]=0 …(4)

For a large number, it is assumed that Ni=Nj and gj is rewritten as gi.

Ni(NlnN)+Ni(iNilngiNi)+Ni(αiNi)Ni(βiNii)=0

The term NlnN is constant with respect to Ni therefore, its derivative with respect to Ni will be zero.

0+(1)lngiNi+Ni(1Ni)+α(1)β(1)i=0lngiNi1+αβi=0

The general expression for the derivative of Φ with respect to ξi and the given equation 17.15 are similar. In both cases, the derivative of a function that contains summation is expressed by the coefficient multiplying variable.

Conclusion

A general expression for the derivative of Φ with respect to ξi is represented as,

δΦδξ1=Ci

Where,

i represents a number that can be either 1,2, or 3.

The general expression for the derivative of Φ with respect to ξi and the given equation 17.15 are similar.

In the given expression it was assumed that for a large system Ni=Nj and gj is rewritten as gi. After the assumption, the equation was derived to get the equation 17.15.

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