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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 is8kBTπm.
  2. The average velocity of gas molecule is zero.

Step by step solution

01

Maxwell Probability Distribution.

P(v)=(m2πkBT)324πv2e-mv2kBT…..(1)

Where,

m is the mass of the particle.

v is velocity of particle.

T is temperature.

kB is Boltzmann constant.

Average speed

vavg=∫0∞vPvdv

Substituting expression (1) in (2).

vavg=∫0∞vm2πkBT324πv2e-mv22kBTdv

Let b=12a2=m2kBT

vavg=4πbπ32∫0∞v3e-bv2dv=4πbπ3212b2=4πb=8kBTπm

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

Nuclear density is approximately 1017 kg/m3. (a) Treating them as a gas of fermions bound together by the (no electrostatic) "strung attraction." calculate EFfor the neutrons in lead-206 (82 protons and 124 neutrons). (b) Treating them the same way. what would EFbe for the protons? (c) In fact, the energies of the most energetic neutrons and protons, those at the Fermi energy, are essentially equal in lead-206? What has been left out of pars (a) or (b) that might account for this?

What information would you need to specify the macro-state of the air in a room? What information would you need to specify the microstate?

Using the result of part (a) in Exercise 74 , determine the number of photons per unit volume in outer space. whose temperature - the so-called cosmic background temperature-is2.7K .

We based the exact probabilities of equation (9-9) on the claim that the number of ways of addingN distinct nonnegative integer quantum numbers to give a total ofM is{M+N-1)!/M!(N-1)!. Verify this claim (a) for the caseN=2,M=5and(b)for the case.

N=5,M=2

To obtain equation (9-42), we calculated a total number of fermions Nas a function of EFassuming T=0. starting with equation(9.41) . But note that (9.4)is the denominator of our model for calculating average particle energy, equation (9.26). its numerator is the total (as opposed (o average particle) energy'. which we’ll callUtotalhere. In other wonts. the total system energy Uis the average particle energyE¯ times the total number of particles (n). CalculateUtotalas a function ofEF

And use this to show that the minimum (T=0)energy of a gas of spin fermions may be written asUtotal=310(3π2ℏ3m3/2V)2/3N5/3
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