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Chapter 9: 42E (page 405)

  1. Calculate the average speed of a gas molecule in a classical ideal gas.
  2. What is the average velocity of a gas molecule?

Short Answer

Expert verified
  1. The average speed of gas molecule in a classical ideal gas is8kBT蟺尘.
  2. The average velocity of gas molecule is zero.

Step by step solution

01

Maxwell Probability Distribution.

P(v)=(m2蟺办BT)324蟺惫2e-mv2kBT鈥�..(1)

Where,

m is the mass of the particle.

v is velocity of particle.

T is temperature.

kB is Boltzmann constant.

Average speed

vavg=鈭�0鈭�vPvdv

Substituting expression (1) in (2).

vavg=鈭�0鈭�vm2蟺办BT324蟺惫2e-mv22kBTdv

Let b=12a2=m2kBT

vavg=4蟺b蟺32鈭�0鈭�v3e-bv2dv=4蟺b蟺3212b2=4蟺产=8kBT蟺尘

02

Average Velocity.

  1. The average velocity of a body is the pace at which it changes position from one location to another. It's a quantity with a vector. The fact that gas molecules move in random directions is well known, and so the gas molecules have velocity in all possible directions. As a result, the vector sum of all velocities equals zero. As a result, a gas molecule's average velocity is zero.

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Most popular questions from this chapter

The temperature of our Sun鈥檚 surface is ~6000K.(a) At what wavelength is the spectral emission of the Sun is maximum? (Refer to Exercise 79.) (b) Is there something conspicuous about this wavelength?

Determine the relative probability of a gas molecule being within a small range of speeds around 2vrmsto being in the same range of speeds around vrms.

Exercise 67 calculates the minimum total energy in a system of spin fermions and is applicable to conduction electrons in a metal. The average particle energy is the total energy divided by the number of particlesN .Show that the average particle energy E鈫�of a conduction electron at low temperature(T=0) is(3/5)EF . This form is convenient, being rather simple, and it can easily he put in terms of N,Vandm via equation .

Derivation of equation(9鈭�40): Our model for calculatingE炉is equation (9-26), whose denominator is the total number of particlesNand whose numerator is the total energy of the system, which we here callUtotal. State with the denominator:

N=鈭�0鈭�N(E)鈭�(E)dE

Insert the quantum gas density of states and an expression for the distribution. using卤to distinguish the Bose-Einstein from the Fermi-Dirac. Then change variables:E=y2, and factorBe+r2/kUTout of the denominator. In the integrand will be a factor

(1鈭�1Be鈭�y2/kBT)鈭�1

Using,(l鈭�蔚)鈭�1鈮�1卤蔚 a sum of two integrals results, each of Gaussian form. The integral thus becomes two terms in powers of1/B. Repeat the process. but instead find an expression forUtotalin terms of1/B, using

U|ntal=鈭�0鈭�EN(E)D(E)dE

Divide your expression forUtotalby that forN. both in terms of1/B. Now1/Bcan safely be eliminated by using the lowest-order expression forNin terms of1/B.

The Fermi velocityVfis defined byEF=12msF2, whereEFis the Fermi energy. The Fermi energy for conduction electron in sodium is 3 IV. (a) Calculate the Fermis velocity (b) What would be the wavelength of an electron with this velocity? (c) If each sodium atom contributes one conduction electron to the electron gas and sodium atom are spaced roughly0.37nmapart. If it is necessary, by the criteria of equation (9-43), to treat the conduction electron gas as quantum gas?
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