Question

what is the proportionality between pressure and temperature? , and the proportionality between atmospheric pressure and temprature?

Answer 1

from the ideal gas equation, PV = nRT
or P = rho*R*T
where P is pressure, R is specific gas constant, rho is density of gas and T is its temeprature
hence
Pressure is directly proportional to temperature for constant density

now, in atmosphere,
lapse rat eis -dT/dh ( h is altitude)
now,
so, dT/dh = k

also, dP/dh = rho*g
hence
dP/dT = rho*g/k
hence

dP = (P/RT)*(g/k)dT
integrating
ln(P) = (g/kR)ln(T)
P = T^(g/kR)

hence pressure is proporitonal to temeprature, but the power is g/kR
whwere k is atmosphjeric lapse rate
R is specific gas constnat
g is acceleration due to gravity