Kinetic-Molecular Theory

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Discussion

Postulates

• all matter is composed of microscopic particles (molecules in general, but also atoms, ions, and free electrons)
• molecules are very small relative to the distance between them
• molecules are in constant random (chaotic) motion
• collisions between particles are perfectly elastic

Macroscopic--Microscopic

• density is the sum of the mass of the molecules divided by the volume that the gas occupies
• pressure is a measure of the mean linear momentum of the molecules
• temperature is a measure of the mean kinetic energy of the molecules

root mean square

[animate]

six modes of freedom

1. heave
2. sway
3. thrust
4. roll
5. pitch
6. yaw

degrees of freedom

• translational: radio frequencies
• rotational: microwave-infrared
• vibrational: infrared-visible-ultraviolet
• electronic: infrared-visible-ultraviolet-x ray

Photons created at the sun's center travel a distance of 2 × 10¹⁰ times the sun's radius before emerging. The trip takes something like 30,000 years.

Summary

• bullet

Problems

practice

1. How about a simple, straightforward problem?
   1. Compute the rms speed of an oxygen molecule at room temperature.
   2. Use the results of part a. to determine the rms speed of a hydrogen molecule at room temperature.
   3. Use the results of part b. to determine the rms speed of a mercury atom at 1200 K.

   Here come the solutions …

   1. Use the formula. Recall that oxygen is a diatomic molecule in everyday situations. Let's not go nuts here with precision. Just use the approximate molecular mass of oxygen (2 × 16 u = 32 u) and the approximate value of room temperature (300 K). Let's try calculating it both ways: first with the mass of a molecule …

      vO = √	⎛ ⎝	3kT	⎞ ⎠	= √	⎛ ⎝	3(1.38 × 10−23 J/K)(300 K)	⎞ ⎠	= 483.5 m/s
      		m				(32 u)(1.66 × 10−27 kg/u)

      and then with the mass of a mole …

      vO = √	⎛ ⎝	3RT	⎞ ⎠	= √	⎛ ⎝	3(8.31 J/mol K)(300 K)	⎞ ⎠	= 483.4 m/s
      		M				(0.032 kg/mol)

      The two answers are slightly different in the fourth significant digit, but I said to be reasonable with the precision. Let's just say the answer is …

      vO = 480 m/s

   2. Exploit the simple ratio of the two molecular masses. Oxygen is 16 times heavier than hydrogen on a per atom or per molecule comparison (since both gases are diatomic in our everyday lives). RMS speed is inversely proportional to the square root of mass (molecular or molar). This means the rms speed of hydrogen should be √16 = 4 times faster. If you would like to see the mathematical reasoning presented formally, here it is …
      
      

      vH

      √

      ⎛ ⎝

      3kT

      ⎞ ⎠

      =

      mH

      = √

      ⎛ ⎝

      mO

      ⎞ ⎠

      = √

      ⎛ ⎝

      32 u

      ⎞ ⎠

      = 4

      vO

      √

      ⎛ ⎝

      3kT

      ⎞ ⎠

      mH

      2 u

      mO

      vH = 4 vO = 4(480 m/s) = 1920 m/s

   3. Another question with rigged numbers. The atomic mass of mercury (200 u) is 100 times that of molecular hydrogen (2 u). This difference reduces the speed by 1/√100 = 1/10. In a similar vein, the temperature of these mercury atoms is 4 times that of the hydrogen molecules in part b. This change raises the rms speed by a factor of √4 = 2. Combining both changes gives a new rms speed that's 2/10 of the old one. Again, if you would like to see the mathematical reasoning presented formally, here it is …
      
      

      vHg

      √

      ⎛ ⎝

      3kTHg

      ⎞ ⎠

      =

      mHg

      = √

      ⎛ ⎝

      THgmH

      ⎞ ⎠

      = √

      ⎛ ⎝

      (1200 K)

      (2 u)

      ⎞ ⎠

      =

      2

      =

      1

      vH

      √

      ⎛ ⎝

      3kTH

      ⎞ ⎠

      THmHg

      (300 K)

      (200 u)

      10

      5

      mH

      vHg = ⅕ vH = 0.2(1920 m/s) = 384 m/s

2. Write something else.
   • Answer it.
3. Write something different.
   • Answer it.
4. Derive the law of Dulong and Petit by applying the equipartition of energy to the atoms in a solid.

   Solution …

   Atoms in a solid have six degrees of freedom. (Why?) Therefore, the energy per atom is

   K =	6	kT = 3kT
   	2

   Multiply by Avogadro's constant to get the internal energy in a mole of atoms.

   U = KNA = 3kTNA

   Specific heat is the rate of change in internal energy with respect to temperature. Molar specific heat is this derivative applied to the internal energy in one mole of atoms.

   CV =	∂	(3kTNA) = 3kNA
   	∂T

   This shows that molar specific heat is a constant for all material since Boltzmann's constant (k) and Avogadro's constant (N_A) are both constant. Substitution using unusually precise values for the two constants yields the usually stated value of this constant.

   CV = 3(1.3806503 × 10−23 J/K)(6.0221415 × 1023 atoms/mol) = 24.94 J/mol·K

numerical

1. What is the average speed of a hydrogen atom in the interplanetary medium (the very thin, cold plasma between the planets) if it has a temperature of 16 K?
2. What is the speed of a typical free electron in copper at room temperature? (Free electrons in a metal are those which are so loosely bound that they wander from atom to atom like the molecules in a gas.)
3. Weapons grade uranium is enriched by the diffusion of gaseous uranium hexafluoride (UF₆) through a stack of very tight metal screens. The lighter, fissionable isotope U²³⁵ has a slightly faster rms speed and is more likely to diffuse through the screens than the heavier, non-fissionable isotope U²³⁸. Determine the ratio of the rms speed of UF₆ made from the light, desirable isotope to the heavier, undesirable isotope. (The atomic mass of fluorine is approximately 19 u.)

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