Question

A marine biologist is preparing a deep-sea submersible for a dive. The sub stores breathing air under high pressure in a spherical air tank that measures 72.0 cm wide. The biologist estimates she will need 5000. L of air for the dive. Calculate the pressure to which this volume of air must be compressed in order to fit into the air tank. Write your answer in atmospheres. Round your answer to 3 significant digits.

Answer 1

To calculate the pressure to which the volume of air must be compressed in order to fit into the air tank, we can use Boyle's law. Given the initial volume of air and the volume of the air tank, we can use the formula P1V1 = P2V2 to solve for the final pressure. The pressure is approximately 20.6 atm.

Step-by-step explanation:

To calculate the pressure to which the volume of air must be compressed in order to fit into the air tank, we can use the formula for pressure-volume relationship known as Boyle's law. Boyle's law states that the pressure of a gas is inversely proportional to its volume at constant temperature. The formula for Boyle's law is P1V1 = P2V2, where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.

In this case, the initial volume is 5000 L and the final volume is the volume of the air tank. Given that the air tank has a width of 72.0 cm, we can calculate its volume using the formula for the volume of a sphere, V = (4/3)πr³, where r is the radius of the sphere. Since the width is given, we need to divide it by 2 to get the radius.

Once we have the initial and final volumes, we can set up the equation:

P1V1 = P2V2

P1(5000) = P2(4/3π(0.72/2)³)

Solving for P2, we get:

P2 = P1V1 / (4/3π(0.72/2)³)

Now we can plug in the values for P1 and calculate P2:

P2 = (1 atm)(5000 L) / (4/3π(0.72/2)³)

Rounding to 3 significant digits, the pressure to which the volume of air must be compressed is approximately 20.6 atm.