nincompoop
  • nincompoop
why is warm air less denser than cool air? mathematical derivation
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • 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?
nincompoop
  • nincompoop
thanks google
anonymous
  • anonymous
lol

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nincompoop
  • nincompoop
moving faster does not mean more space
IrishBoy123
  • IrishBoy123
you could play around with pV = nRT
anonymous
  • anonymous
ok....
nincompoop
  • nincompoop
correct you can use pV=nRT and also use kinetic energy from those moving particles
IrishBoy123
  • IrishBoy123
so \(V \propto T\)
IrishBoy123
  • 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
IrishBoy123
  • IrishBoy123
and then gravity makes it move!!! wind!
nincompoop
  • nincompoop
fun stuff because we have high and low pressures in the atmosphere as well
anonymous
  • anonymous
well you can sub n=m/M
anonymous
  • anonymous
\[PV=\frac{ mRT }{ M }\] \[P=\frac{ mRT }{ VM }\] \[P=\frac{ \rho*RT }{ M }\]
anonymous
  • anonymous
umm
anonymous
  • 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 :)
anonymous
  • anonymous
trying to expand on this..hmm
anonymous
  • anonymous
well we can group M/R since they are simply constants, call it A
anonymous
  • anonymous
\[\rho=\frac{ P }{ AT }\]
anonymous
  • anonymous
so density is inversely proportional to temperature i reckon
anonymous
  • anonymous
so at high Temp, density is low relative to low temp (analogous of cold and hot air)
anonymous
  • 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
anonymous
  • anonymous
and since it pushed up on the cold air, it causes the cold air to be pushed downwards
anonymous
  • anonymous
hope this helps
nincompoop
  • nincompoop
the differential density investigate both in air and in ocean to see how temperature and density interplay.
anonymous
  • anonymous
can we assume all pressure is hydrostatic?
nincompoop
  • nincompoop
will that explain welling effect?
IrishBoy123
  • IrishBoy123
i've just googled welling. never heard of it before. now that is interesting...
IrishBoy123
  • 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!!
anonymous
  • anonymous
hmm, my thermodynamic knowledge may have to come into this...hmm
baewolfstar
  • baewolfstar
Welp I feel stupid
nincompoop
  • nincompoop
https://wiki.met.no/_media/windfarms/brostrom_jms_2008.pdf
anonymous
  • anonymous
that feeling when you get these papers for free at uni. such a gd feeling
anonymous
  • anonymous
nup, im done thinking
anonymous
  • anonymous
my brain will collapse
anonymous
  • anonymous
lel
anonymous
  • anonymous
its an interesting one. we just take this for granted way too easily!
anonymous
  • anonymous
i'm not sure if this help but doesn't hurt looking https://en.wikipedia.org/wiki/Barometric_formula#Density_equations
anonymous
  • anonymous
i can only think that differential density only comes about with changes in altitude...thats why i was thinking hydrostatic pressure
nincompoop
  • nincompoop
correct
nincompoop
  • 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
baewolfstar
  • baewolfstar
Yup still feel stupid
anonymous
  • anonymous
yes
nincompoop
  • nincompoop
thank you, guys I wish I can medal you all
anonymous
  • anonymous
i'd be interested if you could extend my knowledge on this and how we can model it! would be interesting.
anonymous
  • anonymous
haha i can't believe you can distribute owl bucks. thats hectic
anonymous
  • anonymous
good luck!

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