**Concept:** Atomic View of an Ideal Gas

Hey guys, in this video we're going to talk about ideal gases and their atomic view. So what is an ideal gas and what does it look like at the atomic level? Alright let's get to it. Remember that gas is just a type of fluid that will fill the volume of its container just in case you haven't studied phases in forever and don't remember what a gas is. What makes a gas ideal though because ideal gases are very common examples in thermodynamics because they're easy to deal with. An ideal gas has two properties, one that's very important and one that's pretty important but the second one is typically ignored. An ideal gas is a gas whose particles do not interact with one another so they feel no internal forces amongst themselves. The ideal gas approximation works very very well with none charge gases which feel no electric force because they don't have a charge. Additionally the second requirement of an ideal gas is that they're considered to have zero volume. There are plenty of neutral gases that are at very very small atomic sizes that reasonably follow expectations of ideal gases based on certain equations and things that we'll cover. One of the best examples of a near ideal gas is atomic hydrogen which is just a single hydrogen atom. It's going to be the smallest atom so that's the smallest volume so it's the closest to zero and it's uncharged because it's one positive proton and one negative electron so it satisfies those two characteristics of an ideal gas reasonably well. Now an ideal gas is composed of some amount of substance which you can represent either as little N moles or capital N particles. Physicists tend to use capital N particles instead of little N moles which chemists typically use. I'm going to use them both interchangeably because they're both useful. And these particles occupy the container they're in moving in completely random directions at some average speed V. So behind me is an image of a container with an ideal gas inside of it. So these gas molecules are all moving in a completely random directions and they move entirely along the line that they're moving along undeflected because there are absolutely no interactions between these ideal gases. So no interactions mean that there's no reason for them to deflect. They just move and move and move and remember guys that the temperature of this gas is dependent upon the kinetic energy of the particles, the average kinetic energy of the particles don't forget that which is dependent upon V, that average speed that each of these particles have. Now these particles undergo collisions with the wall of the container frequently so they're going to, that one's gonna bounce like this, that one's going to bounce like this, this one's going to bounce. All of these are eventually going to bounce and they make a lot of collisions with the wall. The pressure of a gas which is the force per unit area so the force on one of the walls of the container divided by the area of that wall is dependent on the number of collisions per second with the wall. The idea is the more collisions per second, the higher the pressure. So now let's cover a few conceptual laws about ideal gases. At constant temperature, the smaller the volume of an ideal gas the higher the pressure of an ideal gas. This is called Boyle's Law and it mathematically can be written as P times V equals a constant when V goes down P goes up proportionally. A smaller volume leads to less distance to cross. If you're looking at it from the atomic perspective, let me just just scroll up a little bit here, if we're looking at it from the viewpoint of the gases in that container all the way up a smaller volume means that these gas particles don't have to cross as far of a distance which leads to more collisions per second which means there's a higher pressure. At constant volume the larger the temperature the higher the pressure, this is Charles' Law and it's written as P divided by T equals a constant. So once again they change proportionally. If you double the pressure you double the temperature. This is because at a larger temperature you have a larger average speed. Larger temperature, larger average kinetic energy per molecule, larger average speed that leads to more collisions per second. So just like if you make the volume smaller and there's less of a distance to cross if you keep the volume the same but make them go faster they'll collide with the walls more frequently. There will be more collisions per second and so you will have a higher pressure. Lastly at a constant pressure each particle occupies a constant volume. This is Avogadro's Law. So if you look behind me every single particle has a little volume around it VP I'll call it that each particle occupies and it's the same for every particle. This is not talking about the particle size remember that the individual ideal gas particles occupy no volume, I'm talking about the volume that they occupy by themselves the dedicated volume to them that no other ideal gas particle is in. This is written mathematically as V divided by N equals a constant and that the total number, sorry, the total volume of an ideal gas depends upon the number of particles. In that gas, this is all at constant pressure. That's where Avogadro's Law applies. Let's do a question pertaining to this. An ideal gas at 3 times 10 to the 5 Pascal occupies a volume of half a cubic meter. What volume would the gas occupy if the pressure were doubled at a constant temperature? If the gas contains 5 times 10 to the 24 particles when the volume was half a cubic meter so I'll call this A right here. What volume would the gas occupy if one third of those particles were removed at a constant pressure? I'll call this B. So A and B have two different restrictions. A is at a constant temperature which means that Boyle's Law applies and Boyle's Law is that PV equals a constant. When it's at 3 times 10 to the 5 Pascal, it occupies this volume. What volume would the gas occupy if the pressure were doubled? So if the pressure goes up by two, the volume has to go down by two so that they multiply and still are constant so the volume is halved if the pressure is doubled. This means that the new volume is half the original volume which is 0.25 cubic meters. The original volume was half a cubic meter so the new volume is a quarter of a cubic meter. Part B. Now back when the volume was half a cubic meter, back at our original consideration what if the gas contained 5 times 10 to the 24 particles, what volume would the gas occupy if one third of those particles were removed at constant pressure? Constant pressure means Avogadro's Law. That V over N is a constant. And if N drops down by a factor of 3, sorry N doesn't drop down by a factor of 3, N becomes two thirds N, if you remove one third what you have left is two thirds right? Don't get that confused one third of the particles were removed so you still have two thirds left. That means that the volume has to change in the same way so you have to have two thirds of the volume. If you have two thirds of the gas particles, you have two thirds of the volume. That's what Avogadro's Law says. So the new volume is two thirds the old volume which is 0.17 cubic meters when the old volume is once again half a cubic meter. Alright guys, that wraps up our introduction to ideal gases. Thanks for watching.

**Problem:** A piston is a cylinder with one of the faces free to move. Initially, a piston has a volume of 0.002m^{3} and has air inside of it at atmospheric pressure. If the moving face of the piston is pushed SLOWLY inward, decreasing the volume to 0.00045 m^{3} , what is the final pressure of the air inside of the cylinder?

**Concept:** Ideal Gas Law

Hey guys, in this video we're going to talk about the ideal gas law, it's a law that governs the pressure the volume the amount of particles and the temperature of an ideal gas let's get to it. Combining the three equations we saw previously Avogradro's law Boyle's Law and Charles's law we can formulate what we call the ideal gas law, the ideal gas law is the equation of state for an ideal gas an equation of state is an equation that determines or relates pressure and volume the ideal gas law can be given in two forms in the number of moles and the number of molecules sorry the number of particles like I had said previously in terms of the number of moles its PV equals N, R, T where R is called the ideal gas constant sometimes referred to as a gas constant or the universal gas constant in terms of the particle number. The ideal gas law says PV equals N K T where K is the Boltzmann constant the ideal gas constant is 8.314 in SI units and the Boltzmann constant is 1.38 times 10 to the -23 in SI units. One can easily relate the number of moles to the number of particles in an ideal gas or just in general this isn't actually specific to an ideal gas. You have the number of particles equals the number of moles times a conversion this number is the number of particles per mole. So you're multiplying the number of moles times particles per mole which is why you get the number of particles it turns out that this conversion factor which is called Avogadro's number is actually a constant for all types of particles for all elements across the periodic table it's just 6 times 10 to the 23 and that's a very very important number in chemistry so if any of you guys had taken chemistry before you guys have seen it let's do an example an ideal gas at 1 time 10 to the 5 pascal undergoes a compression at constant temperature if the initial volume is 0.05 cubic meters and the gas is compressed to a 0.01 cubic meters What is the final pressure? So what we want to do is find out what are our Constance? First of all it already tells us temperature is constant, next if the sorry if it's undergoing a compression at a constant temperature just inside of a cylinder inside of a close volume that's compressing there's nothing that implies that particles are actively escaping this gas so we would say that the number of particles is also a constant this means that the ideal gas law PV equals N, K, T actually equals a constant everything on the right hand side of the ideal gas law is a constant because the Boltzmann constants obviously a constant we already said the particle number was a constant and we said the temperature was a constant this if you guys recognize is just Boyle's Law so Boyle's law comes directly from the ideal gas law this means that the pressure times the volume in state one has to equal the pressure times the volume in state two so if we want to solve for the final pressure we just need to divide that volume 2 over to the other side so this is V1 over V2 times P1 which is 0.05 meters cubed that's the initial volume 0.01 cubic meters is the final volume and the initial pressure we're told is atmospheric pressure 1 times 10 to the 5 pascals and this is just 5 times 10 to the 5 pascals. So using the ideal gas law just find out what's a constant come up with something equals a constant and then that something is going to be the same in the initial in the final state and you can solve the problem like that this particular problem happened to just give us Boyle's Law which we already knew but we know that Boyles's, Avogadro's and Charles's law all come from the ideal gas law anyway so it's not surprising that we got Boyle's Law from this. Alright that wraps up our discussion on the ideal gas law. Thanks for watching guys.

**Problem:** A scuba tank is typically filled to a pressure of 3000 psi (2.07 x 10^{7} Pa). When a scuba tank’s pressure gets too high, there is a safety valve that will release all of the air inside before the scuba tank can explode. Let’s say the safety valve on a particular tank will release at 4000 psi. If the tank is filled to 3000 psi with room temperature air (27°C), how hot can the scuba tank get before the safety valve will release?

**Problem:** A tank holds a certain amount of an ideal gas at a pressure of 5.5 x 10^{6} Pa and a temperature of 27°C. If 1/3 of the gas is withdrawn from the tank and the temperature is raised to 50°C, what is the new pressure inside the tank?

A 3.0 L tank contains helium at 3.0 atm and 300 K. The tank is sealed cooled until the pressure is 1.0 atm. What is the final temperature assuming that the volume of the tank was constant?

a) 300 K

b) 200 K

c) 30 K

d) 100 K

e) 150 K

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A metal bulb is filled with 10.2 g of CHCl _{3} and 1.67 g of CH_{4} (two gases). What is the total pressure in the bulb if the temperature is 345°C and the volume is 50.0 mL?

1. 0.107 atm

2. 12.1 atm

3. 193 atm

4. 1450 atm

5. 107 atm

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A certain quantity of a gas occupies 61.3 mL at 68°C. If the pressure remains constant, what would the volume of the gas be at 17°C?

1. 52 mL

2. 92 mL

3. 72 mL

4. 32 mL

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The same number of grams of NH _{3} and O_{2} are placed in separate bulbs of equal volume and temperature under conditions when both gases behave ideally. Which statement is true?

1. The pressure in the NH _{3} bulb is greater than the pressure in the O _{2} bulb.

2. Both bulbs contain the same number of moles of gas.

3. The pressures in the two bulbs are the same.

4. The pressure in the O_{2} bulb is greater than the pressure in the NH _{3} bulb.

5. The bulb containing O_{2} contains more molecules of gas.

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