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 19, Problem 19.73E
Interpretation Introduction

(a)

Interpretation:

The percentage of atoms having velocity within 1% of the root-mean-square speeds from the graph of G(v) is to be stated.

Concept introduction:

The probability distribution function of the velocities of the gas molecules in each dimension is given by gx(vx), gy(vy) and gz(vz). This is a vector quantity. This probability function in the range of to + is represented by,

+gx(vx)dvx=1

The Maxwell-Boltzmann distribution is shown below.

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

This distribution depends on the mass of the particle and absolute temperature.

Expert Solution
Check Mark

Answer to Problem 19.73E

The percentage of atoms having a velocity within 1% of the root-mean-square speeds from the graph of G(v) is approximately 0.068.

Explanation of Solution

The given temperature is 298K.

The molar mass of the helium gas is 4×103kg/mol.

The graph of G(v) verses v is shown in figure 1.

Physical Chemistry, Chapter 19, Problem 19.73E , additional homework tip  1

Figure 1

The formula to calculate the root-mean-square speed is given below.

vrms=3RTM …(1)

Where,

R represents the gas constant with a value of 8.314J/molK.

T represents the temperature.

M represents the molar mass of the gas.

Substitute the values the value of the molar mass of helium, R and T in the equation (1).

vrms=3RTM=(3)(8.314J/molK)(298K)(1kg·m2/s21J)4×103kg/mol=1363.15m/s1363m/s

The corresponding value of the percentage of atoms having 1363m/s velocity is represented as shown below.

Physical Chemistry, Chapter 19, Problem 19.73E , additional homework tip  2

Figure 2

Therefore, the percentage of atoms having velocity within 1% of the root-mean-square speeds from the graph of G(v) is approximately 0.068.

Conclusion

The percentage of atoms having velocity within 1% of the root-mean-square speeds from the graph of G(v) is approximately 0.068.

Interpretation Introduction

(b)

Interpretation:

The percentage of atoms having velocity within 1% of the most probable speed from the graph of G(v) is to be stated.

Concept introduction:

The probability distribution function of the velocities of the gas molecules in each dimension is given by gx(vx), gy(vy) and gz(vz). This is a vector quantity. This probability function in the range of to + is represented by,

+gx(vx)dvx=1

The Maxwell-Boltzmann distribution is given by,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

This distribution depends on the mass of the particle and absolute temperature.

Expert Solution
Check Mark

Answer to Problem 19.73E

The percentage of atoms having velocity within 1% of the most probable speed from the graph of G(v) is approximately 0.072.

Explanation of Solution

The given temperature is 298K.

The molar mass of the helium gas is 4×103kg/mol.

The graph of G(v) verses v is shown in figure 1.

Physical Chemistry, Chapter 19, Problem 19.73E , additional homework tip  3

Figure 1

The formula to calculate the most probable speed is given below as,

νmostprob=2RTM

Where,

R represents the gas constant with a value of 8.314J/molK.

T represents the temperature.

M represents the molar mass of the gas.

Substitute the values the value of the molar mass of helium, R and T in the equation (1).

νmostprob=2RTM=(2)(8.314J/molK)(298K)(1kg·m2/s21J)4×103kg/mol=1113.00m/s

The corresponding value of the percentage of atoms having 1113.00m/s velocity is represented as shown below.

Physical Chemistry, Chapter 19, Problem 19.73E , additional homework tip  4

Figure 3

Therefore, the percentage of atoms having velocity within 1% of the most probable speed from the graph of G(v) is approximately 0.072.

Conclusion

The percentage of atoms having velocity within 1% of the most probable speed from the graph of G(v) is approximately 0.072.

Interpretation Introduction

(c)

Interpretation:

The percentage of atoms having velocity within 1% of the mean speed from the graph of G(v) is to be stated. Whether the corresponding percentages are similar or not is to be stated.

Concept introduction:

The probability distribution function of the velocities of the gas molecules in each dimension is given by gx(vx), gy(vy) and gz(vz). This is a vector quantity. This probability function in the range of to + is represented by,

+gx(vx)dvx=1

The Maxwell-Boltzmann distribution is given by,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

This distribution depends on the mass of the particle and absolute temperature.

Expert Solution
Check Mark

Answer to Problem 19.73E

The percentage of atoms having velocity within 1% of the mean speed from the graph of G(v) is approximately 0.069. All the percentages have a relative same value.

Explanation of Solution

The given temperature is 298K.

The molar mass of the helium gas is 4×103kg/mol.

The graph of G(v) verses v is shown in figure 1.

Physical Chemistry, Chapter 19, Problem 19.73E , additional homework tip  5

Figure 1

The formula to calculate the mean speed is given below as,

ν¯=8RTπM

Where,

R represents the gas constant with a value of 8.314J/molK.

T represents the temperature.

M represents the molar mass of the gas.

Substitute the values the value of molar mass of helium, R and T in the equation (1).

ν¯=8RTπM=(8)(8.314J/molK)(298K)(1kg·m2/s21J)(3.14)(4×103kg/mol)=1256.21m/s1256m/s

The corresponding value of percentage of atoms having 1256m/s velocity is represented as shown below.

Physical Chemistry, Chapter 19, Problem 19.73E , additional homework tip  6

Figure 4

Therefore, the percentage of atoms having velocity within 1% of the mean speed from the graph of G(v) is approximately 0.069.

The percentage of atoms having velocity within 1% of the root-mean-square speeds from the graph of G(v) is approximately 0.068.

The percentage of atoms having velocity within 1% of the most probable speed from the graph of G(v) is approximately 0.072.

Therefore, all the percentages have a relative same value.

Conclusion

The percentage of atoms having velocity within 1% of the mean speed from the graph of G(v) is approximately 0.069. All the percentages have a relative same value.

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