During one of his adventure, Chacha Chaudhary got trapped in an underground cave which was sealed tw — States of Matter and Gaseous State Chemistry Question
Question
During one of his adventure, Chacha Chaudhary got trapped in an underground cave which was sealed two hundred year back. The air inside the cave was poisonous, having some amount of CO (5.0%, by volume) in addition to $O_2$ and $N_2$. 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 exhaled it out in the surroundings. In the meantime, fresh air from surrounding 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 safe Chacha Chaudhary?
💡 Solution & Explanation
Each suck removes half the air, replacing it with fresh air containing 0% CO. The concentration of CO drops by half with each breath. Initial CO is $5.0\%$. Target is $< 0.001\%$. We need $5.0 \times (1/2)^n < 0.001$. $(1/2)^n < 0.001 / 5.0 = 0.0002$. So $2^n > 5000$. Since $2^{12} = 4096$ and $2^{13} = 8192$, it requires 13 breaths to safely lower the concentration below the limit. Formatted as a four-digit integer, it is 0013. Therefore, correct answer is 0013.