Kinetic Molecular Theory Kinetic Molecular Theory (KMT) allows us to explain the observed macroscopic (measurable) properties of ideal gases (pressure, volume, temperature, etc.) in terms of the microscopic properties of individual gas molecules (or atoms, for monatomic gases). As we will see later, real gases do not follow KMT under all conditions, leading to deviations from the ideal gas law. Objectives • Be able to analyze macroscopic properties of gases using postulates of KMT • Calculate ump , urms , uavg for any gas sample •
Sketch Boltzmann distributions for a gas at two different temperatures and explain Model 1: Postulates of the Kinetic Molecular Theory • The particles (molecules or atoms) making up a gas are so small compared to the distances between the particles that the volume of the particles themselves is negligible. In essence, the particles are treated as “point masses” i.e. particles with mass but zero volume. •
The particles move rapidly, in straight lines (unless they undergo a collision) throughout the volume available to them. Not all particles have the same speed; there is a distribution of speeds. •
There are no attractive or repulsive forces between the particles; the particles do not interact at all except during a collision. •
Collisions between a particle and the wall of the container or between particles are perfectly elastic, meaning that the total kinetic energy after the collision is exactly the same as the energy before the collision. •
Pressure is due to collisions of the particles with the walls of the container. •
The average kinetic energy of a collection of gas particles is directly proportional to the Kelvin temperature of the sample. Critical Thinking Questions 1. We have learned that, for an ideal gas, the pressure is directly proportional to the number of moles, if volume and temperature are constant. Explain this in terms of KMT. 2. Many real gases behave fairly ideally under certain conditions; some gases approach ideal behavior more closely than others. Which gas, helium (He) or methane (CH4) do you think would behave most ideally? Explain in terms of KMT. 3. In terms of KMT, why do collisions of gas molecules with the container walls cause pressure? Model 2: The Kinetic Energy of a Particle Suppose we have n moles of an ideal gas in a cubic container with dimension L: Each gas particle is in constant motion, and has mass m. If we consider just a single particle, it has velocity u which has components in the x, y, and z directions ux, uy, and uz such that u 2 = u x2 + u y2 + uz2
It can be shown (this derivation is shown in your textbook in great detail; you won’t need to reproduce it!) that 1
mu 2
1
P = nN A
or PV = nN A mu 2
3
V
3
Eq. 1 Where u 2 is the average velocity of a gas particle (since there are a huge number of gas particles, with many different velocities). Recall that the kinetic energy of a single particle is ½ mu2, and the kinetic energy of one mole of particles is NA (½ mu2) so our equation can be written as follows: 2 ⎤
1
⎡
2 ⎢ nN A 2 mu ⎥
Eq. 2 P=
⎥
3⎢
V
⎣
⎦
(
Since KE
avg
= NA
And finally, since (
1
2
)
)
mu 2 we can modify this equation as follows: P=
2 ⎡ n KE
⎢
3⎣
V
avg
⎤
PV 2
= KE
⎥ or
n
3
⎦
avg
Eq. 3 KMT tells us that the kinetic energy of an ideal gas is directly proportional to absolute temperature, we can write: PV
Eq. 4 ∝ KE avg ∝ TK n
Thus, using KMT (only) we have derived an expression that is essentially equivalent to the ideal gas law! Finally, we can combine the ideal gas law with the above expression to obtain an equation for the average kinetic energy of an ideal gas: PV
2
= RT = KE avg
n
3
Eq. 5 3
KE avg = RT
2
In this equation, we must use R=8.31 J/K-­‐mol. Eq. 5 essentially tells us the definition of absolute temperature of a gas: it is a measure of the motion of the particles. Critical Thinking Questions 4. According to the ideal gas law (PV=nRT) does the quantity PV depend on the identity of the gas at a given temperature? (i.e. does it matter whether the gas is He, Ar, methane?) Explain. 5. According to equation (1) does the quantity PV depend on the identity of a gas at a given temperature? Explain. 6. Given the equation KE
avg
= NA
(
1
2
)
mu 2 does the quantity on the right depend on the identity of a gas at a given temperature? Explain. 7. One of the postulates of the KMT says that the kinetic energy of an ideal gas depends only on the absolute temperature. Thus, one mole of He has exactly the same kinetic energy as one mole of methane at the same temperature, even though the mass of He is much lower than the mass of methane. How is this consistent with your answers to numbers 1-­‐3? Explain. 8. Tank A contains H2(g) and Tank B (which is identical to Tank A) contains the same number of moles of He(g). Both tanks are at the same temperature, and have the same volume. Assume that both gases obey the ideal gas law. a. Which tank (if either) has the higher pressure? Explain. b. In which tank (if either) is the average kinetic energy of the gas particles greater? Explain. c. In which tank (if either) is the speed of the gas particles greater? Explain. Model 3: The Speeds of Particles Within a sample of a gas (at constant temperature), there is a distribution of speeds; some particles move relatively slowly while others move very rapidly. The distribution function for gas speeds was developed by James Clerk Maxwell in 1860 but was later influenced by Ludwig Boltzmann, and is now called the Maxwell-­‐
Boltzmann distribution function. It is given by the following: 3
mu 2
⎛ m ⎞ 2 2 − 2 kBT
f (u) = 4π ⎜
u e
⎝ 2k BT ⎟⎠
where: u = velocity in m/s m = mass of a gas particle in kg kB = Boltzmann’s constant = 1.38066 × 10-­‐23 J/K T = temperature in K Eq. 6 A plot of the M-­‐B equation for N2 at STP is shown here: On this plot, we can see that the “most probable” velocity for N2 at STP is around 400 m/s. The equation for the most probably velocity is given by: 2kBT
2RT
u
=
=
mp
Eq. 7 m
M
Critical Thinking Questions 9. In Eq. 7, what is the mathematical relationship between kB and R? 10. At 273 K, the most probably velocity for N2(g) is around 400 m/s. Would the most probably velocity at 1000K be higher, lower, or the same? Explain. 11. On the plot for O2 below (where T=273K) sketch a line corresponding to what you think the distribution would look like at a temperature of 1000 K (you may extend the x-­‐axis if you want). Do not continue to the next question until you have done this one. 12. Below is a graph of the M-­‐B distribution for N2 at three different temperatures. Notice that as the temperature increases, the average velocity increases, and the width of the distribution increases (the area under all three curves is the same). 13. Draw a similar sketch to the above for He, Ne, Ar, and Xe, all at 273 K. The graph can be qualitative, but be sure that the relative average speeds make sense! 14.