Physics for Scientists and Engineers with Modern Physics
Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 20, Problem 39AP

(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 39AP

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

Explanation of Solution

Given information: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.

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.

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)

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 .

Conclusion:

Therefore, 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 39AP

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

Explanation of Solution

Given information: Value of average speed is vmp10.0 .

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

Nv(v)Nv(vmp)=(vvmp)2e(1(vvmp)2)

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 .

Conclusion:

Therefore, 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 39AP

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

Explanation of Solution

Given information: Value of average speed is vmp2.00 .

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

Nv(v)Nv(vmp)=(vvmp)2e(1(vvmp)2)

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

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

Conclusion:

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 39AP

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

Explanation of Solution

Given information: Value of average speed is vmp .

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

Nv(v)Nv(vmp)=(vvmp)2e(1(vvmp)2)

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

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

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

Conclusion:

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 39AP

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

Explanation of Solution

Given information: Value of average speed is 2.00vmp .

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

Nv(v)Nv(vmp)=(vvmp)2e(1(vvmp)2)

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 .

Conclusion:

Therefore, 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 39AP

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

Explanation of Solution

Given information: Value of average speed is 10.0vmp .

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

Nv(v)Nv(vmp)=(vvmp)2e(1(vvmp)2)

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 .

Conclusion:

Therefore, thenumerical 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 39AP

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

Explanation of Solution

Given information: Value of average speed is 50.0vmp .

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

Nv(v)Nv(vmp)=(vvmp)2e(1(vvmp)2)

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 .

Conclusion:

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

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