Q21 of 24 Page 75

(a) Pressure decreases as one ascends the atmosphere. If the density of air is ρ, what is the change in pressure dp over a 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 p0.


(c) If p0 = 1.03×105 N m-2, ρ0 = 1.29 kg m-3 and g = 9.8 m s-2, at what height will the pressure drop to (1/10) the value at the surface of the earth?


(d) This model of the atmosphere works for relatively small distances. Identify the underlying assumption that limits the model.


(a) As we go up in the atmosphere, pressure decreases.


Consider a layer of atmosphere at height h and area A.


Let the pressure at h be p + dp.


Let the pressure at h + dh be p.



This layer is in equilibrium and hence, upward and downward forces are balanced.


Pressure × Area = Force = ma =




Negative sign indicates that pressure decreases with height.


(b) Here,


Hence,






Integrating on both sides,





(c)




(d) We can assume only under isothermal conditions which is valid near the surface of the earth and not at great heights.


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If a drop of liquid breaks into smaller droplets, it results in lowering of temperature of the droplets. Let a drop of radius R, break into N small droplets each of radius r. Estimate the drop-in temperature.

 20

The surface tension and vapour pressure of water at 20°C is 7.28×10-2 Nm-1 and 2.33×103 Pa, respectively. What is the radius of the smallest spherical water droplet which can form without evaporating at 20°C?

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Surface tension is exhibited by liquids due to force of attraction between molecules of the liquid. The surface tension decreases with increase in temperature and vanishes at boiling point. Given that the latent heat of vaporisation for water Lv = 540 k Cal kg-1, the mechanical equivalent of heat J = 4.2 J cal-1, density of water ρ w = 103 kg l –1, Avogadro’s No NA = 6.0 × 1026 k mole –1 and the molecular weight of water MA = 18 kg for 1 k mole.

(a) estimate the energy required for one molecule of water to evaporate.


(b) show that the inter–molecular distance for water is []1/3 and find its value.


(c) 1 g of water in the vapor state at 1 atm occupies 1601cm3. Estimate the intermolecular distance at boiling point, in the vapour state.


(d) During vaporisation a molecule overcomes a force F, assumed constant, to go from an inter-molecular distance d to d′. Estimate the value of F.


(e) Calculate F/d, which is a measure of the surface tension.


 23

A hot air balloon is a sphere of radius 8 m. The air inside is at a temperature of 60°C. How large a mass can the balloon lift when the outside temperature is 20°C? (Assume air is an ideal gas, R = 8.314 J mole -1 K-1, 1 atm. = 1.013×105 Pa; the membrane tension is 5 N m-1.)