Physics for Scientists and Engineers, Technology Update (No access codes included)
Physics for Scientists and Engineers, Technology Update (No access codes included)
9th Edition
ISBN: 9781305116399
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Question
Chapter 21, Problem 21.67AP

(a)

To determine

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp50.0.

(a)

Expert Solution
Check Mark

Answer to Problem 21.67AP

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp50.0 is 1.09×103.

Explanation of Solution

Value of average speed is vmp50.0.

Write the expression for the Maxwell-Boltzmann speed distribution function,

    Nv=4πN(m02πkBT)32v2e(m0v22kBT)                            (1)

Here, Nv is the Maxwell-Boltzmann speed distribution function, N is the total number of molecules of gas, T is the absolute temperature of gas, v is the speed of the fraction of molecules of gas, kB is the Boltzmann constant and m0 is the mass of the gas molecule.

Write the expression for the average speed of a gas molecule.

    v=8kBTπm0

Here, v is the average speed of a gas molecule.

Write the expression for the most probable speed of a gas molecule.

    vmp=2kBTm0

Here, vmp is the most probable speed of a gas molecule.

Write the formula to calculate the numerical value of the Nv(v)Nv(vmp) using equation(1).

    Nv(v)Nv(vmp)=4πN(m02πkBT)32v2e(m0v22kBT)4πN(m02πkBT)32vmp2e(m0vmp22kBT)=(vvmp)2e(m0vmp22kBTm0v22kBT)=(vvmp)2em0vmp22kBT(1(vvmp)2) (2)

Substitute 2kBTm0 for vmp in equation (2) to find Nv(v)Nv(vmp).

    Nv(v)Nv(vmp)=(vvmp)2em0(2kBTm0)22kBT(1(vvmp)2)=(vvmp)2e(1(vvmp)2)                                                  (3)

Conclusion:

Substitute vmp50.0 for v in equation (3) to find Nv(v)Nv(vmp),

    Nv(v)Nv(vmp)=(vmp50.0vmp)2e(1(vmp50.0vmp)2)=1.0868×1031.09×103

Thus, the numerical value of the Nv(v)Nv(vmp) for the value of v=vmp50.0 is 1.09×103.

(b)

To determine

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp10.0.

(b)

Expert Solution
Check Mark

Answer to Problem 21.67AP

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp10.0 is 2.69×102.

Explanation of Solution

 Value of average speed is vmp10.0.

From equation (3), Write the formula to calculate the numerical value of the Nv(v)Nv(vmp)

    Nv(v)Nv(vmp)=(vvmp)2e(1(vvmp)2)

Conclusion:

Substitute vmp10.0 for v in above expression to find Nv(v)Nv(vmp).

    Nv(v)Nv(vmp)=(vmp10.0vmp)2e(1(vmp10.0vmp)2)=2.69×102

Thus, the numerical value of the Nv(v)Nv(vmp) for the value of v=vmp10.0 is 2.69×102.

(c)

To determine

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp2.00.

(c)

Expert Solution
Check Mark

Answer to Problem 21.67AP

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp2.00 is 0.529.

Explanation of Solution

 Value of average speed is vmp2.00.

Recall equation (3)

Conclusion:

Substitute vmp2.00 for v in equation (3) to find Nv(v)Nv(vmp).

    Nv(v)Nv(vmp)=(vmp2.00vmp)2e(1(vmp2.00vmp)2)=0.529

Therefore, the numerical value of the Nv(v)Nv(vmp) for the value of v=vmp2.00 is 0.529.

(d)

To determine

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp.

(d)

Expert Solution
Check Mark

Answer to Problem 21.67AP

The numerical value of the Nv(v)Nv(vmp) for the value of v=vmp is 1.00.

Explanation of Solution

 Value of average speed is vmp.

Recall equation (3)

Conclusion:

Substitute vmp for v in above expression to find Nv(v)Nv(vmp).

    Nv(v)Nv(vmp)=(vmpvmp)2e(1(vmpvmp)2)=1.00

Therefore, the numerical value of the Nv(v)Nv(vmp) for the value of v=vmp is 1.00.

(e)

To determine

The numerical value of the Nv(v)Nv(vmp) for the value of v=2.00vmp.

(e)

Expert Solution
Check Mark

Answer to Problem 21.67AP

The numerical value of the Nv(v)Nv(vmp) for the value of v=2.00vmp is 0.199.

Explanation of Solution

 Value of average speed is 2.00vmp.

Recall equation (3)

Conclusion:

Substitute 2.00vmp for v in above expression to find Nv(v)Nv(vmp).

    Nv(v)Nv(vmp)=(2.00vmpvmp)2e(1(2.00vmpvmp)2)=0.199

Thus, the numerical value of the Nv(v)Nv(vmp) for the value of v=2.00vmp is 0.199.

(f)

To determine

The numerical value of the Nv(v)Nv(vmp) for the value of v=10.0vmp.

(f)

Expert Solution
Check Mark

Answer to Problem 21.67AP

The numerical value of the Nv(v)Nv(vmp) for the value of v=10.0vmp is 1.01×1041.

Explanation of Solution

 Value of average speed is 10.0vmp.

Recall equation (3),

Conclusion:

Substitute 10.0vmp for v in above expression to find Nv(v)Nv(vmp).

    Nv(v)Nv(vmp)=(10.0vmpvmp)2e(1(10.0vmpvmp)2)=1.01×1041

Thus, the numerical value of the Nv(v)Nv(vmp) for the value of v=10.0vmp is 1.01×1041.

(g)

To determine

The numerical value of the Nv(v)Nv(vmp) for the value of v=50.0vmp.

(g)

Expert Solution
Check Mark

Answer to Problem 21.67AP

The numerical value of the Nv(v)Nv(vmp) for the value of v=50.0vmp is 1.25×101082

Explanation of Solution

Reacall equation (3)

Conclusion:

Substitute 50.0vmp for v in above equation to find Nv(v)Nv(vmp).

    Nv(v)Nv(vmp)=(50.0vmpvmp)2e(1(50.0vmpvmp)2)=1.25×101082

Thus, the numerical value of the Nv(v)Nv(vmp) for the value of v=50.0vmp is 1.25×101082.

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Chapter 21 Solutions

Physics for Scientists and Engineers, Technology Update (No access codes included)

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