Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Textbook Question
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Chapter 4, Problem 4.36E

Which of the following functions are exact differentials?

(a) d F = 1 x d x + 1 y d y

(b) d F = 1 y d x + 1 x d y

(c) d F = 2 x 2 y 2 d x + 3 x 3 y 3 d y

(d) d F = 2 x 2 y 3 d x + 2 x 3 y 2 d y

(e) d F = x n d x + y n d y , n = any integer

(f) d F = ( x 3 cos y ) d x + ( x 3 sin y ) d y

Expert Solution
Check Mark
Interpretation Introduction

(a)

Interpretation:

Whether the function, dF=1xdx+1ydy is an exact differential or not is to be stated.

Concept introduction:

A function is said to be an exact differential if it satisfies the relation given below.

(x(Fy)x)y=(y(Fx)y)x

In the above equation, F is the function having variables x and y.

Answer to Problem 4.36E

The given function, dF=1xdx+1ydy is an exact differential.

Explanation of Solution

The given function is written below.

dF=1xdx+1ydy

Using the identity of exact differential, the function can be differentiated as shown below.

(Fy)x=1y2(x(Fy)x)y=0(Fx)y=1x2(y(Fx)y)x=0

Since, (x(Fy)x)y=(y(Fx)y)x for the given function. Therefore, it is an exact differential.

Conclusion

The given function, dF=1xdx+1ydy is an exact differential.

Expert Solution
Check Mark
Interpretation Introduction

(b)

Interpretation:

Whether the function, dF=1ydx+1xdy is an exact differential or not is to be stated.

Concept introduction:

A function is said to be an exact differential if it satisfies the relation given below.

(x(Fy)x)y=(y(Fx)y)x

In the above equation, F is the function having variables x and y.

Answer to Problem 4.36E

The given function, dF=1ydx+1xdy is not an exact differential.

Explanation of Solution

The given function is written below.

dF=1ydx+1xdy

Using the identity of exact differential, the function can be differentiated as shown below.

(Fy)x=1x(x(Fy)x)y=1x2(Fx)y=1y(y(Fx)y)x=1y2

Since, (x(Fy)x)y(y(Fx)y)x for the given function. Therefore, it is not an exact differential.

Conclusion

The given function, dF=1ydx+1xdy is not an exact differential.

Expert Solution
Check Mark
Interpretation Introduction

(c)

Interpretation:

Whether the function, dF=2x2y2dx+3x3y3dy is an exact differential or not is to be stated.

Concept introduction:

A function is said to be an exact differential if it satisfies the relation given below.

(x(Fy)x)y=(y(Fx)y)x

In the above equation, F is the function having variables x and y.

Answer to Problem 4.36E

The given function, dF=2x2y2dx+3x3y3dy is not an exact differential.

Explanation of Solution

The given function is written below.

dF=2x2y2dx+3x3y3dy

Using the identity of exact differential, the function can be differentiated as shown below.

(Fy)x=9x3y2(x(Fy)x)y=27x2y2(Fx)y=4xy2(y(Fx)y)x=8xy

Since, (x(Fy)x)y(y(Fx)y)x for the given function. Therefore, it is not an exact differential.

Conclusion

The given function, dF=2x2y2dx+3x3y3dy is not an exact differential.

Expert Solution
Check Mark
Interpretation Introduction

(d)

Interpretation:

Whether the function, dF=2x2y3dx+2x3y2dy is an exact differential or not is to be stated.

Concept introduction:

A function is said to be an exact differential if it satisfies the relation given below.

(x(Fy)x)y=(y(Fx)y)x

In the above equation, F is the function having variables x and y.

Answer to Problem 4.36E

The given function, dF=2x2y3dx+2x3y2dy is not an exact differential.

Explanation of Solution

The given function is written below.

dF=2x2y3dx+2x3y2dy

Using the identity of exact differential, the function can be differentiated as shown below.

(Fy)x=4x3y(x(Fy)x)y=12x2y(Fx)y=4xy3(y(Fx)y)x=12xy2

Since, (x(Fy)x)y(y(Fx)y)x for the given function. Therefore, it is not an exact differential.

Conclusion

The given function, dF=2x2y3dx+2x3y2dy is not an exact differential.

Expert Solution
Check Mark
Interpretation Introduction

(e)

Interpretation:

Whether the function, dF=xndx+yndy is an exact differential or not is to be stated.

Concept introduction:

A function is said to be an exact differential if it satisfies the relation given below.

(x(Fy)x)y=(y(Fx)y)x

In the above equation, F is the function having variables x and y.

Answer to Problem 4.36E

The given function, dF=xndx+yndy is an exact differential.

Explanation of Solution

The given function is written below.

dF=xndx+yndy

Using the identity of exact differential, the function can be differentiated as shown below.

(Fy)x=nyn1(x(Fy)x)y=0(Fx)y=nxn1(y(Fx)y)x=0

Since, (x(Fy)x)y=(y(Fx)y)x for the given function. Therefore, it is an exact differential.

Conclusion

The given function, dF=xndx+yndy is an exact differential.

Expert Solution
Check Mark
Interpretation Introduction

(f)

Interpretation:

Whether the function, dF=(x3cosy)dx+(x3siny)dy is an exact differential or not is to be stated.

Concept introduction:

A function is said to be an exact differential if it satisfies the relation given below.

(x(Fy)x)y=(y(Fx)y)x

In the above equation, F is the function having variables x and y.

Answer to Problem 4.36E

The given function, dF=(x3cosy)dx+(x3siny)dy is not an exact differential.

Explanation of Solution

The given function is written below.

dF=(x3cosy)dx+(x3siny)dy

Using the identity of exact differential, the function can be differentiated as shown below.

(Fy)x=x3cosy(x(Fy)x)y=3x2cosy(Fx)y=3x2cosy(y(Fx)y)x=3x2siny

Since, (x(Fy)x)y(y(Fx)y)x for the given function. Therefore, it is not an exact differential.

Conclusion

The given function, dF=(x3cosy)dx+(x3siny)dy is not an exact differential.

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