## nincompoop one year ago why is warm air less denser than cool air? mathematical derivation

1. anonymous

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

2. nincompoop

3. anonymous

lol

4. nincompoop

moving faster does not mean more space

5. IrishBoy123

you could play around with pV = nRT

6. anonymous

ok....

7. nincompoop

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

8. IrishBoy123

so $$V \propto T$$

9. IrishBoy123

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

10. IrishBoy123

and then gravity makes it move!!! wind!

11. nincompoop

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

12. anonymous

well you can sub n=m/M

13. anonymous

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

14. anonymous

umm

15. anonymous

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 :)

16. anonymous

trying to expand on this..hmm

17. anonymous

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

18. anonymous

$\rho=\frac{ P }{ AT }$

19. anonymous

so density is inversely proportional to temperature i reckon

20. anonymous

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

21. anonymous

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

22. anonymous

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

23. anonymous

hope this helps

24. nincompoop

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

25. anonymous

can we assume all pressure is hydrostatic?

26. nincompoop

will that explain welling effect?

27. IrishBoy123

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

28. IrishBoy123

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!!

29. anonymous

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

30. baewolfstar

Welp I feel stupid

31. nincompoop
32. anonymous

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

33. anonymous

nup, im done thinking

34. anonymous

my brain will collapse

35. anonymous

lel

36. anonymous

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

37. anonymous

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

38. anonymous

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

39. nincompoop

correct

40. nincompoop

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

41. baewolfstar

Yup still feel stupid

42. anonymous

yes

43. nincompoop

thank you, guys I wish I can medal you all

44. anonymous

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

45. anonymous

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

46. anonymous

good luck!