Today we talked about work and the expansion and compression of gasses. Prof. Mason started with a set up like the one showing below. It consisted of a low-friction syringe that was connected to a rubber tube. The latter was connected to a bulb through a one-hole stopper, and the whole system was closed. There were also beakers with cool and hot water. Only one of the beakers is shown in the picture below.
The picture above was taken the moment after Prof. Mason compressed the syringe to show that it takes energy to compress a gas. This can be called doing work on the gas. He also talked about how spending energy to compress the gas ended up giving energy to the gas as it was compressed and increased the pressure and temperature of the gas. When the volume was reduced, the gas particles collided more times with the walls. This is an increase in pressure. And, with more collisions, the particles moved faster. This is an increase in temperature. Quantitatively the energy that Prof. Mason spent in compressing was gained by the gas.
Then, he submerged the bulb in cool water and let his students see the the syringe move down. The moving down of the syringe indicates a reduction of volume of the gas inside. This means that the gas was putting less pressure which means that they weren't pushing as hard against the walls of the syringe. This means that they had less energy. This was absorbed by the cool water.
Then, Prof. Mason submerged the bulb in hot water and let his students see the syringe move upwards, indicating an increase of volume. It seems like the hot water warmed the bulb and the gas inside. When the gas inside was warmed, it increased in volume. This is because the heat flowed from the water to the gas and made the gas particles move rapidly and push harder against the walls of the syringe.
Next we calculated the change in internal energy done by a solid with a very low expanding coefficient.
First we calculated the work done by it as shown in the picture above.
Next we explored the kinetic theory of molecular motion. In the demo, we observed the movement of two particles inside a box and saw how they collided with the walls and each other. We used this model to see how the gas law and kinetic theory think about particles.
They have the following assumptions. Every collision is completely elastic. Molecules are point like, i.e. it assumes molecules are made of only one atom. It also assumes that they are mostly empty space and have negligible mass.
When we apply these assumptions to a lot of molecules, it looks like the picture above. This means that the gas law and kinetic theory work effectively for very limited scenarios such as molecules made up of only one gas particle.
We used these assumptions to come up with formulas that we can work with that can yield measurable results.
We start with imagining an empty box of length X. A particle takes sometime to reach one wall when coming from the opposite side. We can calculate its velocity by dividing the length of the box by the time it took to travel from side to side. We do this for the x, y and z directions, and obtain the total velocity to be the square root of 3 times the velocity along the x direction. It was assumed that the velocities along every direction were equal. Then, we calculate the force on the molecule by the wall once it changes direction. Since force is equal to the change in momentum divided by the time it takes, we calculated the momentum of the particle before and after the collision and calculate the change and divide it by the time it took to do so. This yields the force to be the mass times the velocity squared divided by the length of the cube.
Then we calculated the pressure by this particle using P = F/A and multiply it by n to expand our applicability of the formula to a gas with n moles of particles. We also transformed it to have a relationship between pressure and kinetic energy of the molecule.
Since pressure times volume can be expressed in several ways, relationships between this ways can be obtained. Our new relationship allows us to connect vrms to temperature
Next we studied a couple of processes: isothermal and adiabatic.
When a process is occurring on a gas and there is no change in temperature, all of the heat that is added or subtracted goes towards work. This yields the formulas on the bottom left of the above picture. It is also depicted that this process must be slow to allow the temperature to remain the same. Meanwhile, when a process is occurs on a gas and there is no heat that is absorbed or released, the expansion or contraction of a gas simply changes the internal energy of it. Hence we obtain the formulas on the bottom right of the above picture.
In an adiabatic process, there is no change in pressure and no heat flow interaction with the outside. Hence, the change in internal energy is equal to the work done by the gas. This equation can be reduced to the top formula on the above picture. When the expression is integrated, we obtained the bottom right expression that is true for adiabatic processes.
Next we used this formula to calculate the temperature that the inside of tube below would reach and see if it corroborated with our observations.
In the video above, we put a small piece of cotton inside a tube whose volume could be reduced quickly. Our expectation was that the quick compression of the air inside the tube would make the temperature rise and ignite the cotton. This happened wonderfully and we could observe some smoke coming out. Our calculations using the formula above showed that the final temperature inside the tube was 373.15 K. This means that cotton can be ignited at about this temperature. If instead we would've put a piece of paper inside, it wouldn't have ignited since its flash point is 451 F or about 505.93 K and this temperature wasn't reached inside the tube.
Summary
Today we talked about the work done on or by gases and saw their compression and expansion as a function of temperature. Then we calculated the change in internal energy for a solid with a very small coefficient of thermal expansion and saw that the work done by it can be negligible. Next we developed the kinetic theory and saw all of its assumptions and limitations. We developed the relationship between the root-mean-square speed and temperature of a gas, provided that they follow the ideal gas law. And finally, we developed some of the formulas for a process to be isothermal or adiabatic.
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