14.2 - Kinetic Model of Gases Flashcards
What can we assume about molecule movement and volume?
Molecules don’t interact at all and volume occupied is negligible. All collisions are assumed to be perfectly elastic where no energy is lost. This not only applies for molecule to container collisions but inter molecular collisions. Volume occupied is negligible, so we assume collision frequency between molecules is too.
Describe the simple model of one molecule in a box and the impulse formula.
A molecule of mass m is in a box of length x, height y, and depth z. It travels at speed c,which replaces v for ideal gases. Assuming all collisions are elastic, the molecule bounces in the x direction with velocity changing from +c to -c. The change in momentum for collisions with the wall is +/- 2mc.
What is the formula for impulse, and what can we learn about time and distance travelled from this?
Impulse = F * change in t. Average F=2mc/change in t. Time is the time between one collision on the wall, and the next one. Distance moved in this time is 2x.
What’s the formula for force we gain from the change in time formula?
c=2x/t. change in t=2x/c. F=2mc/2x/c=2mc^2/2x=mc^2/x
Describe the model of molecule movement for a more realistic model : N molecules in a box.
A typical volume of a gas contains more than one molecule of gas. with N molecules in a box, 1/3 move in the y direction, 1/3 in x, and 1/3 in the z direction. Alternatively, adding all vector components of velocities will give the same result.
What’s the force formula for N molecules in a box?
F= N/3 * mc^2/x = Nmc^2/3x
How can the pressure formula be gained from the force formula for N molecules in a box?
Pressure=F/A=Nmc^2/3x / yz= Nmc^2/3xyz = Nmc^2/3V
Describe the full ideal gas model, N molecules with varying speeds in a box, and why the mean square is needed.
Molecules not only travel in random directions, but with varying speeds. pV=1/3 Nmc^2 should use an average. This isn’t average speed, as F=mc^2/x, c^2 is an important measure of force, so this should be averaged using the mean square… č ^2
What’s the formula for pV using the mean square c.
pV=1/3 Nmč^2
How do Van Der Waals forces limit ideal behaviour in a box model?
As a gas molecule approaches the edge of a box, it experiences Van der Waals forces which will pull it away from the box, reducing momentum, according to Newton’s third law. Pressure on the wall will rise. Gases which are well separated display ideal behaviour.
How is the formula 1/2mč^2=3/2kT derived? What does it show?
Mc^2 is part of the ideal gas equation, looking very similar to average kinetic energy of a molecule. As pV=1/3Nmč^2, this equals 2/3N(1/3mč^2). As pV=NkT, NkT=2/3N(1/2mč^2). 1/2mč^2=3/2kT. Mean kinetic energy of all molecules in a mixture of air is the same at a given temperature. Larger molecules such as nitrogen will travel slower than smaller molecules, they have less rms speed.
What tells us molecules are bound to collide, and what affects distance travelled between collisions?
It’s assumed molecules in ideal gas occupy negligible volume but they do have finite size and as many molecules move in random directions, they’re bound to frequently collide.
On average each air molecule travels 100nm between collisions. Under the same temperature and pressure, larger molecules travel a smaller distance than smaller collisions.
Why is it hard to predict collision paths?
As each collision changes an atoms’ direction, it’s hard to predict the path. Even the longest path, 500 steps, takes less than 1ms to complete.
Graphs with x and y axes showing number of mean free paths can show collision.
Given a molecule’s molar mass, mean free path, and temperature, alongside displacement from bottle, how can the time for the particle to reach this be found?
- Fulfil 3/2kT=1/2Nmč^2. Find the mass using m=molar mass/Avogadro constant, and find temperature in Kelvins to obtain rms.
- The displacement from the starting point is equal to the mean free path, path per step, multiplied by the square root of the number of steps, square root N.
- N is found by squaring this answer, and is multiplied by the mean free path to find the actual distance travelled. Divide this by rms to obtain time.
In reality time will be much less, this assumes the experiment occurs in a test tube where there are no air draughts or sudden movement.
