(a). It is known that density `rho` of air decreases with height y as
`rho=rho_(0)e^(-y//y(_o))` ltb rgt where `p_(o)=1.25kgm^(-3)` is the density at sea level, and `y_(o)` is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant
(b). A large He balloon of volume `1425m^(3)` is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise?
[take `y_(o)=8000m and rho_(He)=0.18kgm^(-3)`]

Explanation

We know that the rate of decrease of density `rho` of air with height y is directly proportional to density `rho` i.e.,
`-(d rho)/(dy) prop rho or (dp)/(dy) =-krho`
where `K` is a constant of proportional . Here -ve sign shows that `rho` decreases as y increase.
`:. (d rho)/(rho)=-K dy`
Intergrating it within constions , as `y` changes from `0` to `y` , density change from `rho_(0)` to `rho`, we have
`int_(rho_0)^(rho) (d rho)/(rho) = -int_(0)^(y) K dy`
`[ log_(e)rho]_(rho_0)^(rho) = -Ky` or `log_(e)rho-log_(e)rho_(0)=-Ky`
or `log(rho)/(rho_0) = -Ky` or `(rho)/(rho_0) = e^(-Ky)` or `rho=rho_(0)e^(-Ky)`
Here `K` is a constant . suppose `y_(0)` is a constant such that `K=1//y_(0)`, then `rho=rho_(0)e^(-y//y_(0))`.
(b) The balloon will rise to a height, where its density becomes equal to the air at that height.
Density of balloon, `rho = (mass)/(volume) = ("pay load" + "mass of He")/("volume") = ((400+1425xx0.18)kg)/(1425m^(3)) = (656.5)/(1425) kg//m^(3)`
As, `rho = rho_(0)e^(-y//y_(0)) :. (656.5)/(1425) = 1.25e^(-y//8000)`
or,`e^(-y//8000) = (656.5)/(1425xx1.25) or e^(y//8000) = (1425xx1.25)/(656.5) = 2.7132`
Taking log on both the sides `(y)/(8000) = log_(e)2.7132=2.3026 log_(10)2.7132 = 2.3026xx0.4335 ~~1`
`y=8000xx1 = 8000m = 8km`.