During one of his adventures, Chacha Chaudhary got trapped in an underground cave which was sealed two hundred years back. The air inside the cave was poisonous, having some amount of CO (5.0%, by vol

Question

During one of his adventures, Chacha Chaudhary got trapped in an underground cave which was sealed two hundred years back. The air inside the cave was poisonous, having some amount of CO (5.0%, by volume) in addition to O2 and N2. The safe limit of CO in the atmosphere is less than 0.001%, by volume. Sabu, being huge, could not enter into the cave, so in order to save Chacha Chaudhary, he started sucking the poisonous air out of the cave, by mouth. Each time, he fills his lungs with the cave air and exhales it out in the surroundings. In the meantime, fresh air from surroundings effused into the cave till the pressure becomes again 1 atm. If each time, Sabu sucked out half of the air present in the cave, how many times does Sabu need to suck out air in order to save Chacha Chaudhary?

Answer:

13.00

Solution:

Each suck removes half the air; fresh (CO-free) air then refills the cave to 1 atm, so the CO mole fraction is halved per cycle: c_n = 5.0% x (1/2)ⁿ. Safe condition: 5.0 × (1/2)ⁿ < 0.001, i.e. 2ⁿ > 5000. Since 2¹² = 4096 < 5000 and 2¹³ = 8192 > 5000, Sabu must suck n = 13 times. (After 13 cycles, c = 5/8192 = 6.1 × 10^-4 % < 0.001%.)

TestHub · Accepted Answer

Each suck removes half the air; fresh (CO-free) air then refills the cave to 1 atm, so the CO mole fraction is halved per cycle: c n = 5.0% x (1/2) n . Safe condition: 5.0 × (1/2) n < 0.001, i.e. 2 n > 5000. Since 2 12 = 4096 < 5000 and 2 13 = 8192 > 5000, Sabu must suck n = 13 times. (After 13 cycles, c = 5/8192 = 6.1 × 10 -4 % < 0.001%.)

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