Home
Class 11
PHYSICS
Assume that the temperature remains esse...

Assume that the temperature remains essentially constant in the upper part of the atmosphere. Obrain an epression for the variation in pressure in the upper atmosphere with height, the mean molecular weight of air is M.

Text Solution

Verified by Experts

Suppose the pressure at height h is p and that at `h+dh is p+dp. Then `
`dp=-rhog dh. …(i)`
Now considering any small volume `DeltaV` if air of mass
`Deltam, `
`pDeltaV=nRt=(Deltam)/(M)RT`
`p=(Deltam)/(DeltaV) (RT)/(M)=(rhoRT)/(M)`
`or, =rho=(M)/(RT)p.`
Putting in (i),
`dp=-(M)/(RT)pg dh`ltbr.`or, `int_(p_(0))^(p)(dp)/(p)=int_(0)^(h)-(M)/(RT)g dh`
`or, In(p)/(p_(0))=-(Mgh)/(RT)`
where p_(0) is the pressure at h=0.`
Thus, `p=p_(0) e^(-(Mgh)/(RT)).`
Promotional Banner

Similar Questions

Explore conceptually related problems

It is known that density rho of air decreases with height y as rho=rho_0e^(-y//y_0) where rho_0=1.25 kg m^-3 is the density at sea level. And y_0 is a constant . This density variation is called the law of atmosphere. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant. A large He balloon of volume 1425 m^3 is used to lift a payload of 400kg.Assume that the balloon maintains constant radius as it rises. How high does it rise?

It is known that density rho of air decreases with height y as -y//y_(0) rho=rho_(o)e where rho_(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 ? (y_(o)=8000mandrho_(He)=0.18kgm^(-3)) .

(a) Pressure decreases as one ascends the atmosphere . If the density of air is rho ,what is the change in pressure dp over differential height dh ? (b) Considering the pressure P to be proportional to the density find the pressure P at a height h if the pressure on the surface of the earth is P_(o) . (c ) If P_(o)=1.03xx10^(5)N//m^(-2),rho_(o)=1.29kg//m^(3)andg=9.8m//s^(2) what height will the pressure frop to (1)/(10) the value at the surface of earth ? (d) This model of the atmosphere works for relatively small distance .Identify the underlying assumption that limits the model.

A wind with speed 40m//s blows parallel to the roof of a house .The area of the roof is 250m^(2) . Assuming that the pressure inside the house is atmospheric pressure ,the force exerted by the wind on the roof and the direction of the force will be (rho_(air)=1.2kg//m^(3))

The helicopter has a mass m and maintains its height by imparting a downward momentum to a column of air defined by the slipstream boundary as shown in figure. The propeller blades can project a downward air speed v_0 , where the pressure in the stream below the blades is atmospheric and the radius of the circular cross section of the slipstream is r . Neglect any rotational energy of the air, the temperature rise due to air friction and any change in air density rho . If the power is doubled, the acceleration of the helicopter is :-

The helicopter has a mass m and maintains its height by imparting a downward momentum to a column of air defined by the slipstream boundary as shown in figure. The propeller blades can project a downward air speed v, where the pressure in the stram below the blades is atmospheric and the radius of the circular cross section of the slipstream is r. Neglect any rotational energy of the air, the temperature rise due to air friction and any change in air density rho . If the power is doubled, the acceleration of the helicopter is :-

A cylindrical block of length 0.4 m and area of cross-section 0.04 m^2 is placed coaxially on a thin metal disc of mass 0.4 kg and of the same cross - section. The upper face of the cylinder is maintained at a constant temperature of 400 K and the initial temperature of the disc is 300K . if the thermal conductivity of the material of the cylinder is 10 "watt"// m.K and the specific heat of the material of the disc is 600J//kg.K , how long will it take for the temperature of the disc to increase to 350 K ? Assume for purpose of calculation the thermal conductivity of the disc to be very high and the system to be thermally insulated except for the upper face of the cylinder.

A small spherical monoatomic ideal gas bubble (gamma=5//3) is trapped inside a liquid of density rho (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T_0 , the height of the liquid is H and the atmospheric pressure P_0 (Neglect surface tension). The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant)

A small spherical monoatomic ideal gas bubble (gamma= (5)/(3)) is trapped inside a liquid of density rho_(l) (see figure) . Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T_(0) , the height of the liquid is H and the atmospheric pressure is P_(0) (Neglect surface tension). When the gas bubble is at a height y from the bottom , its temperature is :-

A small spherical monoatomic ideal gas bubble (gamma=5//3) is trapped inside a liquid of density rho (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T_0 , the height of the liquid is H and the atmospheric pressure P_0 (Neglect surface tension). As the bubble moves upwards, besides the buoyancy force the following forces are acting on it