Consider a rising parcel of air. Before rising the temperature, pressure and den

Question

Consider a rising parcel of air. Before rising the temperature, pressure and density of the air parcel are T1, p1, p1. After rising the temperature, pressure and density of the air parcel are T2, p2, p2. a) If the pressure of the air parcel decreases by a factor of 2 (p2/p1 = 0.5) while the temperature remains constant (T2 = T1), what is the change in density? Please express your answer as he ratio p2/p1. b) In actual fact, the temperature of a rising parcel always decreases (T2 < T1). Express the ratio p2/p1, which is density in terms of T1 and T2. Is your ratio p2/p1 larger or smaller than the ratio found for part a?

Explanation / Answer

Considering air to be an ideal gas

Using ideal gas relation

P1V1 / T1 = P2V2 / T2

where P = pressure

T = Temperature

V = Volume

Dividing above by mass of the air M gives us

P1/T1(M/V1) = P2/T2(M/V2)

Density p = M/V

So

P1/T1p1 = P2/T2p2

p2/p1 = (P2/P1)*(T1/T2)

a. For P2/P1 = 0.5

T2/T1 = 1

p2/p1 = 0.5*1 = 0.5

b. For P2/P1 = 0.5

Since T2<T1

T1/T2 > 1

So

p2/p1 = (0.5)*(T1/T2)

Since, the ratio T1/T2 >1, p2/p1 shall be > 0.5

So, the ratio is larger

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