Kinetic Molecular Theory (KMT) Notes

Kinetic Molecular Theory (KMT)

Kinetic Molecular Theory (KMT)

• A model that attempts to explain gas behavior.

• Postulates of the KMT:

  • Gases are mostly empty space; the volume of the particles is negligible.
  • Gas particles are in constant random motion. Pressure is due to the collisions of the gas particles with the container walls.
  • Gas particles neither attract nor repel each other.
  • The average kinetic energy of a gas sample is proportional to the Kelvin temperature.

  $KE_{AVE} = (3/2)RT$ where $R = 8.3145 J/K \cdot mol$

Path of One Particle in a Gas

• Gases are in constant random motion.
• $Pressure = \frac{Force \ of \ Collisions}{Area}$

Boyle's Law

• $PV = constant$
• at constant n and T
• Volume is decreased.

Avogadro's Law

• $V = constant(n)$
• at constant P and T
• Gas cylinder- Moles of gas increase.
• Increase volume to return to original pressure.

P vs n

• at constant V and T

Kinetic Energy of Gases

• $KE = energy \ due \ to \ motion: KE = \frac{1}{2}mv^2$ where m = mass and v = velocity
• From KMT, average KE directly proportional to T: $(KE)_{Ave} = (3/2)RT$ where $R = 8.3145 J/K \cdot mol$

Average Kinetic Energy

• $(KE)_{Ave} = (3/2)RT$ and $KE = \frac{1}{2}mv^2$\ *
• Why talk about the average KE? Molecules possess many different KE’s; easiest to talk about average.
• Why do we have a distribution of KE? Molecules possess many different velocities at a specific temperature.
• Why is $(KE)_{Ave}$ directly proportional to T? As T increases, the average velocity of the gas molecules increases, which causes the average KE to increase.

Velocity Distribution at STP for an $O_2$ Gas Sample

• Relative number of $O_2$ molecules with given velocity.
• Molecular velocity (m/s)

Velocity Distributions at Different Temps

• Relative number of $N_2$ molecules with given velocity

Charles's Law

• $V = constant(T)$
• at constant n and P
• Temperature is increased.

P vs T

• at constant n and V
• Temperature is increased.

Relative Velocities of Gas Particles

• Compare the average velocity of a sample of $H<em>2$ molecules to a sample of $O</em>2$ molecules at some constant temperature? Which has the faster average velocity, or do the two different gas samples have the same average velocity?
• Answer: the sample of the smaller $H_2$ molecules must have the faster average velocity. In general, at constant temperature, the lighter the gas molecules, the faster the average velocity.

Relative Molecular Speed Distribution of $H<em>2$ and $UF</em>6$

• Percentage of molecules.

Question on Velocity Distributions

• If the gases represented in the following plot are He, $H<em>2O$, $N</em>2$, $O<em>2$, and $H</em>2$, which curve represents the molecular speed distribution for $N_2(g)$?

KMT Question

• Consider two gases, A and B, in identical containers of equal volume. Both containers are at the same temperature and pressure.

  • A: mass = 0.34 g
  • B: mass = 0.48 g

• How many of the following five statements are true?

  • I. The number of molecules of A is equal to the number of molecules of B. True
  • II. The molar mass of A is greater than the molar mass of B. False
  • III. Both samples have the same average kinetic energy. True
  • IV. The molecules of A have the same average velocity as the molecules of B. False
  • V. The molecules of A collide with the container walls more frequently than the molecules of B. True

Relative Velocities Question

• How much faster, on average, are $H<em>2$ molecules as compared to $O</em>2$ molecules (at some nonzero Kelvin temperature)?

Calculating Relative Velocities

• Graham (1830) – postulated that the average velocity of a molecule is inversely proportional to the square root of the molar mass (M).
• $v = average \ velocity = \sqrt{\frac{3RT}{M}}$
• For two gases:
  $\frac{v<em>1}{v</em>2} = \sqrt{\frac{M<em>2}{M</em>1}}$

Measuring Relative Velocities

• Diffusion - rate that gases mix
• Effusion - rate that gases pass through a tiny hole
• Both quantities are directly proportional to the average velocity of the gas.

Effusion of Gas into Evacuated Chamber

Diffusion Rates of $NH_3$ and $HCl$ Molecules Through Air

• $NH<em>3(g) + HCl(g) \rightarrow NH</em>4Cl(s)$

Real Gases

• Some assumptions of the kinetic molecular theory are oversimplifications of real gas properties.
• The two assumptions that don’t always hold true for real gases are:
  • Gas particles do experience intermolecular attractions. Effect? P that we measure (real) is less than the P if the gas is ideal.
  • Gas particles have a real volume that may not be negligible. Effect? V of a container (real) is not the actual V of empty space available for the gas particles to move about.

Effect of Intermolecular Forces

VAN DER WAALS EQUATION FOR REAL GASES

• “a” and “b” are constants that are experimentally determined for each gas.
• “a” is directly related to the strength of the intermolecular forces.
• “b” is directly related to molecular size.

When Does a “Real” Gas Behave Most “Ideal”?

• The best conditions for ideal behavior are at high temperatures and low pressures. Why?
  • At high temperatures, the gas particles are moving very fast. This minimizes the effect of intermolecular forces between gas particles.
  • At low pressure, there is a lot of empty space between gas particle. This minimizes the effect of volume of gas particles.

Low Pressure vs High Pressure

• Low Pressure - lots of space between molecules. Volume of container taken up by the volume of the molecules themselves is minimal.
• Higher Pressure – gas particles are more crowded. The volume of the container taken up by the volume of the molecules themselves is much more pronounced.