Question

why is warm air less denser than cool air?
mathematical derivation

Answer 1

because the molecules in warm air move faster than cool air, creating more space for every molecule... i think... is that what you need?

Answer 2

thanks google

Answer 3

lol

Answer 4

moving faster does not mean more space

Answer 5

you could play around with pV = nRT

Answer 6

ok....

Answer 7

correct you can use pV=nRT and also use kinetic energy from those moving particles

Answer 8

so \(V \propto T\)

Answer 9

so if the air is in a confied space it retains its density but pressure increases

in atmosphere air can move - pressure makes it move, so volume etc etc

Answer 10

and then gravity makes it move!!! wind!

Answer 11

fun stuff because we have high and low pressures in the atmosphere as well

Answer 12

well you can sub n=m/M

Answer 13

\[PV=\frac{ mRT }{ M }\]
\[P=\frac{ mRT }{ VM }\]
\[P=\frac{ \rho*RT }{ M }\]

Answer 14

umm

Answer 15

Assuming it is a ideal gas you can use the ideal gas equation and the definition of molar mass in the following way.

First consider the ideal gas equation
\[pV=RnT\]
And substitute in molar amount from the definition of molar mass
\[n=\frac{ m }{ M }\]
This gives you
\[pV=\frac{ RmT }{ M }\]
Now you can isolate \[m/V\] which is the density
\[\rho =\frac{ m }{ V }=\frac{ pM }{ TR }\]
As you can see the density is inversely proportional to the temperature.
Hopes this answers your question :)

Answer 16

trying to expand on this..hmm

Answer 17

well we can group M/R since they are simply constants, call it A

Answer 18

\[\rho=\frac{ P }{ AT }\]

Answer 19

so density is inversely proportional to temperature i reckon

Answer 20

so at high Temp, density is low relative to low temp (analogous of cold and hot air)

Answer 21

plus, when people usually say that hot air rises, it is actually false. it depends on the system parameters.

Using the nomenclature that hot air pushes up on the cold air would be better

Answer 22

and since it pushed up on the cold air, it causes the cold air to be pushed downwards

Answer 23

hope this helps

Answer 24

the differential density
investigate both in air and in ocean to see how temperature and density interplay.

Answer 25

can we assume all pressure is hydrostatic?

Answer 26

will that explain welling effect?

Answer 27

i've just googled welling. never heard of it before. now that is interesting...

Answer 28

is that part of life on earth? stuff dredged up by wind and the coriolis force, add a dash of sunshine, keep repeating, and 3 billions years later,... the iPad!!

Answer 29

hmm, my thermodynamic knowledge may have to come into this...hmm

Answer 30

Welp I feel stupid

Answer 31

https://wiki.met.no/_media/windfarms/brostrom_jms_2008.pdf

Answer 32

that feeling when you get these papers for free at uni. such a gd feeling

Answer 33

nup, im done thinking

Answer 34

my brain will collapse

Answer 35

lel

Answer 36

its an interesting one. we just take this for granted way too easily!

Answer 37

i'm not sure if this help but doesn't hurt looking
https://en.wikipedia.org/wiki/Barometric_formula#Density_equations

Answer 38

i can only think that differential density only comes about with changes in altitude...thats why i was thinking hydrostatic pressure

Answer 39

correct

Answer 40

the rule of thumb on less dense being on top much like oil and water also apply with cool air and warm air when talking about air or air pockets and the atmosphere

Answer 41

Yup still feel stupid

Answer 42

yes

Answer 43

thank you, guys
I wish I can medal you all

Answer 44

i'd be interested if you could extend my knowledge on this and how we can model it! would be interesting.

Answer 45

haha i can't believe you can distribute owl bucks. thats hectic

Answer 46

good luck!