Particulate to Macroscopic Transformations

The postulates of KMT define the behavior of the particles of the gas, but the gas laws are empirical laws, in that they are derived from experimental measurements, which are not at the particulate (molecular) level. This means they are based on macroscopic observations of many molecules and not the behavior of one. A liter of oxygen is an easily measurable quantity, but how many molecules would be in a liter at STP?.. and would you expect them all to have the same energy?


Exercise \(\PageIndex{1}\)

How many particles of Xenon are in 1.00 liter at STP?

Answer

STP is 1 bar (0.98592 atm) at 273.15K, which is the freezing point of water, and we know that the molar volume of any ideal gas is 22.4L. So: \<1.00L(\frac{1molXe}{22.4L})\frac{6.022x10^{23}atoms}{mol}=2.69x10^{22}atoms\; Xe\>


Yes, that was 2.693x1022 atoms in one liter! So when we describe a gas we are not describing a molecule, but an ensemble of molecules, a group of particles that is observed as the whole, not at the level of the individual molecule. This means there is a distribution of characteristics of the individual gas particle, like velocity. At any point in time some molecules are moving real fast, others slow, and some may in fact be in the middle of a collision at are not moving at all, but changing the direction that they are moving in.

*

Figure \(\PageIndex{1}\): On the left is an illustration of gas particles in a container and on the right is the velocity profile for a gaseous system The discrete particles that form the gas have a range of velocities, and their distribution over the range is expressed by the area under the curve.

Typically a curve like the one in Figure \(\PageIndex{1}\) represents a macroscopically observable system in the order of a mole of particles. At any given time there is a distribution of velocities as described by the curve, and the measured values of macroscopic observables like pressure and temperature are all related to the shape of this curve, and likewise, if those values change, the shape of the curve changes.


Exercise \(\PageIndex{2}\)

What is wrong with the label of the x-axis of the velocity distribution curve in Figure \(\PageIndex{1}\)?

Answer

Velocity is a vector and has direction, which means that the distribution of speeds is mapped out in 3D space (left side of Figure \(\PageIndex{1}\), and so although some gasses may be moving at 400 m/s in one direction, other molecules will be moving the same speed in the opposite direction with a cancelling out effect. Speed is the magnitude of velocity, and the x-axis really represents the distribution of the speed of the particles in the ensemble.


Although the kinetic energy of each particle in a stationary gas is 1/2mv2, we can not use the average velocity of the particles because it vectoraly adds to zero. Instead we relate its kinetic energy to the room mean square velocity, \(\mu_{rms}\).


\(\mu_{rms}\)

For a gaseous system the average kinetic energy is

\

Average root mean square speed for a system of n particles with individual velocities vi:\<\mu_\ce{rms}=\sqrt{\overline{v^2}}=\sqrt{\dfrac{v^2_1+v^2_2+v^2_3+v^2_4+…}{n}}\>


The important take-home message of Figure 10.5.1 is that a gas is a system of particles, each particle of which has a unique location, velocity and kinetic energy, and that we use \(\mu_{rms}^2\) to calculate the average kinetic energy of the gas (even though most of the particles actually have a different energy).

\

It can be shown that \(E_{K_{ave}}\) of a collection of gas molecules is directly proportional to the temperature of the gas and may be described by the equation:






Boyle"s Law & KMT (const. n,T)

At constant n and T, the velocity profile on part (b) of Figure 10.5.1 stays the same. Here we are looking at closed (constant n) isothermal (constant T) elastic system that has no resistance or heat loss to expansion or contraction.

a. Effect of \(\Delta V \) on P:Reducing the volume of the cube in part (a) of Figure 10.5.1 increases the pressure because it increases the collision frequency of the particles with the wall. This is because they are maintaining the same velocity but have less distance to travel as they traverse the cube. Likewise, Increasing Volume decreases pressure beacuse it takes longer to traverse the expanded cube, resulting in a reduction in the collision frequency. Remember Pressure is force per unit area, and the force is the consequence of the change in momentum of the gas particle as it undergoes an elastic collision with the wall of the container (this is called an impulse in physics).

b. Effect of \(\Delta P\) on V:

Figure 10.5.2 shows this effect. Here we have a cylinder type system with a gas in a closed system. If the internal pressure is not equal to the external, the system will adjust until the are equal. Consider a system starting at state (b) where the internal volume is 3 dm3 and the pressure is 1.97 atm. If you then reduced the external pressure to 0.987 atm, the force/unit ares inside is greater than outside and the system expands until they balance and equilibrium is re-established (a). Note, in this case we cut the pressure in half, and so the volume doubled.

*

Can you cool the curve to absolute zero?

No, at some point you reach the boiling point and cooling below that causes the system to convert to a liquid, and you no longer have a gas.


Gay-Lussac"s Law (constant n,V)

This is similar to Charles Law, except now instead of having a perfectly elastic container (that has no resistance to expansion or contraction) you have a rigidcontainer of constant size. As T goes up the root mean velocity goes up and so the average kinetic energy goes up and thus the force imparted by collisions with the container are greater. Also, since the molecules tend to be moving faster (remember, there is a velocity profile), the collision frequency is greater.


Avogadro"s Law (const P,T)

This describes an elastic container at isobaric and isothermal conditions. According to KMT increasing the number of particles increases the collision frequency on the surface which effectively increases the force. Since pressure (force per unit area) is constant, the system must expand (increase surface area) to accommodate the increased number of collisions.

Animation 10.5.1: Change the number of molecules and click the start button. Note how adding molecules increases the collision frequency. Since pressure is being kept equal, the system adjusts its volume so the collision frequency returns to the original value.