10.2 Gas Laws (including the ideal gas law)

Question 1

Ideal Gas Law

To describe any system of a gas you’ve really only got 4 variables to work with
1) Pressure (P)
2) Volume (V)
3) Number of moles of the gas (n)
4) Temperature (T)

In a multivariable system like this you’re going to have to hold 2 of the variables constant while manipulating the other 2.
* for instance if you’re going to compare pressure to volume (like Boyle’s law), you’re going to have to hold # of moles of the gas and temp constant.
* This is kind of implied when working through some of these equations

Question 2

Boyle’s Law
* He says that pressure is inversly proportional to volume
* As pressure goes up, volume goes down, and as volume goes up, pressure goes down
* think pushing on a balloon so much that it pops. as the pressure went up the volume went down
* also just think about just pushing on a soda can, the more pressure you put on it the smaller its volume gets.

Charles Law
* Compares volume to temp
* says they’re directly proportional as long as the other variables are held constant
* So if volume goes up, temperature goes up, and if volume goes down, temperature goes down
* So think about having a balloon inside a cool house, to outside during a hot summer. Well this balloon will actaully expand when you take it outside, meaning as temperature increased, the volume increased as well
* Think about the bag of chips at snowshoe. When we brought it inside from outside the cool mountain to inside the warm house it also expanded.

Avogardro’s Law
* Volume is proportional to # of moles of gas
* as you add more moles of gas, the volume of that gas is going to incrase (duh), assuming the other 2 variables are held cosntant
* Think about blowing up a balloon, as you blow it up you put more moles of gas in it, and at the same time it gets bigger = increased volume.

Keep in mind w/ the equations below, they’re all dependent on the other 2 vairables being held constant.

Isothermal = the same temperature or constant temperature
* if we carry out a process isotheramlly for a specific sample of gas - so were not adding or substracting gas (n held constant) and isothermal = temperature held constant. So this means 2/4 of the variables are held constant.
* under these conditions is where pressure and volume are inversely proportional.

Question 3

Charles Law

V is proportional to T
* Temperature is always in kelvin in this chapter
* C does not have a true 0
* K is an absolute 0 in the universe. You can’t go colder than this. Its the complete adbsence of heat. You can’t have a lower temperature than that.
* They might get tricky and put this in C, must be in K

V/T = a constant
* Well if this ratio always gave you the same constant, if you double the temperature in kelvin, then you’d also have to double the volume so that it = the same exact constant (has to be the same ratio to = the same constant)
* Another way to express this is V1/T1 = V2/T2
* If they give us 3/4 of the conditions were going to have to solve for the 4th

so below I know it must be 2L because if I doubled the K than the L must double to have the same ratio so it spits out the same constant.

Question 4

Avogardo’s Law
* Volume is proportional to the # of moles of gas as long as pressure and temperature are held constant

Another way to say this is that V/N = a constant
* again, if you double the moles of gas, you’d have to double the volume to keep that same constant.

V1/N1 = V2/N2

Question 5

Explaining these equations

V is proportional to T
* that just means when temperature goes up, volume goes up

V/T = k
* k = a constant #
* so basically no matter what the volume or temperature are, the ratio between them stays the same.
* If volume = 2L and temperature = 300 K, then V/T = 2/300 = 0.0067
* Later, if volume = 3 L, then T must be 450 K to keep that same ratio: 3 / 450K = 0.0067.
* So the trick is that they increase at the same proportion to keep that constant

V1/T1 = V2/T2
* if the ratio is always a constant

V1/T1 = K and V1/T2 = K
* than both of those ratios equal the same number so

V1/T1 = V2/T2

If the car goes 60 miles in 1 hour, then later 120 miles in 2 hours, the ratio is the same

60/1 = 120/2 = 60 mph
* so the ratio remains constant so they = the same thing
* so either of them = 60, so they must be = to each other

EX: A balloon has a bolume of 2.0 L at a temperature of 300 K. If the temperature increases to 450 K, what is the new volume of the balloon.

V1 = 2.0 L
T1 = 300 K
T2 = 450 K
V2 = ?

2.0/300 = V2/450

Solve for V2 = 3.0 L

So, as the temperature increases from 300K to 450 K, the volume increases from 2.0 L to 3.0 L. The volume expanded because the temeprature went up.

Since V1/T1 has a constant ratio that leads to a constant, than I can set it to V2/T2 because its technicaly the exact same number because the constant is the same, meaning the ratio is also the same. So if they’re = than I can set them = to eachother, which is where the equation V1/T1 = V2/T2 comes from. Its kind of like saying 1/1 = 1/1, because each are the exact same ratio, thereby yielding the same constant

Question 6

Boyels law

p is proportional to 1/v
* meaning they’re inversely proportional
* if volume increases pressure decreases, and vice versa - as long as the temp and the amount of gas stay constant

p = k/v
* can turn this into pv = k

as volume changes, pressure changes in a way taht keeps their product constant (they have a constant, and as 1 increases the other decreases to keep that constant constant)

why we change the 1 to a k:
* K is some number, it an be 1, 5 , 100 anything depending on the gas you’re dealthing with

Imagine you’re baking and the recipe says
* The amount of sugar is proportional to the number of cookies
* 10 cookies –> 1 cup of sugar
* 20 cookies –> 2 cups of sugar
* thats a proportional relationship

But you cant say sugar = cookies
* because that would mean 10 cookies = 10 cups of sugar

Instead you write sugar = k x cookies
* here k - 0.1 because 10 cookies = 1 cup of sugar

when you says P is proportional to 1/v
* you’re saying pressure on 1/v, but the exact value of pressure needs a multiplier
* so we write p = k/v
* because that 1 was never meant to be a fixed 1. It just represented “some number” that keeps the proportioanlity true. and that sume number is K

Question 7

Combined gas law * P1V1/N1T1 = P2V2 / N2T2

Question 8

The reason we use K:

Because kelvin is an absolute temperature scale meaning
* 0 K = absolute zero. The point where all particle motion stops

Why celsius doesnt work
* If you use celsius, you could accidentally divide by zero or even get negative values, which physically makes no sense for things like volume (because remember, volume is directly proportional to temperature)

Question 9

Temp in kelvin

as long as you use the same units on both sides for pressure and volume, it doesnt matter which units you choose (its a ratio, they’ll cancel eachother out if you use them on both sides)
* Temp units matter because we cannot get a negative number or a 0 for it, must be in kelvin, would blow up the entire equation

0C = 273K

What is the final volume of the gas below
* notice n jsut cancels because we didnt change the sample at all
* You can see that the pressure doubled - we know that pressure and volume are inversly proportional - so id I doubled the pressure, I must have to half the volume to keep these two equations = to each other
* This makes a ton of sense looking at the equation, if the number ontop doubled (1ATM –> 2ATM), then the other # ontop must be cut in half because those 2 things = eachother, so things must be held constant
* Temperature on the bottom, doubled, and we learn if we double the temperature, the volume doubles (directly proportional), meaning that the volume ontop should be doubled.
* So this essentially cancels out meaning, we don’t change the volume at all
* so V1 = 1L, and so does V2

Question 10

When to use the ideal gas low vs combined gas law

Combined: P1V1 / N1T1 = P2V2 / N2T2

Ideal gas law: Pv = NRT

So we use the ideal gas law when I have one set of conditions (like not V1 and V2, just a singlular v)

Question 11

Ideal gas law: called this because only a gas thats behaving ideally will follow this law
* not all gases exhibit ideal behavior
* Ideal gas doesnt actually exist - however, there are conditions where gases are more likely to behave ideally
* gases most likely to behave ideally under condotions of low pressures and high temperatures (which means they’ll follow are equation better)

PV = NRT

This is all part of kinetic molecule theory
1) Gas molecules have negligible volume - most of a gas is made of empty space - under conditions of low pressure. The percentage of volume of a gas taken up by the little molecules themselves is negligbile (its almost all empty space)
2) Gas molecules don’t have any attractive forces between themselves - it turns out all molecules are a little bit sticky, however its so negligible that were going to assume its 0 - however, this statment is most true at high temperatures (they’re the least sticky). This is because at higher temperature molecules are moving faster. When molecules are moving fast, that means they collide faster and bounce away faster, giving them less overall time to experience those attractive (sticky) forces.
3) All collisions are elastic (no loss of kinetic energy) - when the molecules hit eachother / other things in the environment, we assume exactly no loss of kinetic E (even though thats not entirely true) - often molecules lose some kinetic E because they have the propensity to stick to eachother, and they use that kientic force to overcome that sticking, however, in the real world it would use some of that kinetic E, thereby slowing it down (however, were assuming it retains all that kinetic E, because were assuming those molecules arent going to stick together, even though, again, this is untrue).
4) A gas is composed of a large number atoms/molecules in a random motion.
5) KE average is proportional to T. - if you double the temperature of a gas, you double its average kinetic energy.

should memorize all these, but the first 2 are the most important.

NOTE: we need all these things to hold true for PV = NRT to be accurate
* This equation lets us describe
* the above make a gas ideal, however, this doesnt actually happen.
* Low pressure / high temperature, makes this equation most true

Question 12

Ideal Gas law:
* PV = nRT

R = universal gas constant
* 0.08206 Latm / molk - used more when the problem is talking about mols (much more commonly used one)
* Or 8.314 J/mol*k - used more when the equation is talking about E

EX: What is the volume of 2 moles of argon gas behaving ideally at a temperature of 298K and have a pressure of 2.0 atm.
* So in this problem it doesnt matter that its argon, at all, Completely irrelevent.
* Temp is already in kelvin, which is what we need
* R is always a constant

Question 13

What is the volume of 80g of argon gas behaving ideally at a temperature of 25 C and a pressure of 1520 torr.
* The problem is that all the units are messed up

have to know that 1 ATM = 760 torr

Question 14

RMS speed
* This is root, mean squared speed

You talk about a RMS when theres no overall value for that particular quantitify before its going all directions.
* So think about the gas molecules around you, they have just has a high likelyhood as going 1 direction as another. But as a net they have no direction
* so if we calculated the average velocity it would be 0
* thats not very helpful to figure out how fast the gas molecules in the air are actually moving and thats where RMS comes in.

RMS: kind of like an average when you dont want direction to cancel them all out and give you an average value of you.

Need everything in SI units so that it works out in m/s
* That means M (molar mass) needs to be in Kg - because for no fucking reason the Kg is the unit of mass instead of g
* also, this is really the only time well use 8.314 J/mol*K for R

Question 15

What is the root mean square speed of O2 gas molecules at 273K

U = speed

So its asking for a velocity in m/s, so need to make sure entire thing is in SI unts

temp must be in K

M = molar mass for oxygen = 16, however its talking about O2 so its 32g/mol
* however, i need kg/mol, not grams. because we need it in SI units

461 m/s
* so this tells us the average speed of 1 of the O2 molecules in this room is 461 m/s
* the units are weird, and dont really seem to cancel

Question 16

maxwell distribution of speeds

Red line:
* as you get higher temperatures you get higher Urms

Green = highest temperature (because it has the highest velocity on average, followed by blue then red)
* can tell be that Urms equation from the picture below

You could also look at it as molar mases
* increased molar mass (M) = lower velocity (shown in the equation below)
* So the green line must have the lowest molar mass, followed by the blue then the red

Question 17

Answer 17

An increase in the frequency of molecules colliding w/ the walls of the container
* With the same # of molecules in a smaller volume there will be more collisions with the walls of the container. As the temperature remains constant the average speed of the molecules will not change and therefore the average force of the collisions of molecules w/ the walls of the walls of the container will not change either.

Question 18

Answer 18

So the pressure of the gas is pushing down on the left side and the pressure of the ATM is pressing down on the R side.

You can see the gas is pressing w/ a higher pressure (because the gas has pushed more of the liquid over to the ATM side)

So the difference in pressure between the two sides (delta P) = 19cm (w/ the atm side not pushing as much)
* 19cmHg = 190mmHg
* 1 ATM = 760 mmHg
* so we know the pressure on the gas side is 190mmHg greater than the ATM side
* 190 + 760 = 950 mmHg

So the gas is pushing down w/ 950 mmHg of pressure

Question 19

Answer 19

Question 20

Answer 20

I just used the equation pv = nrt and saw that pressure and temperature werent inversely proportional

Question 21

Answer 21

just worked from pv=nrt

Question 22

Answer 22

Question 23

Answer 23

assume baseline is all 1’s