Questions: 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 70.0 cm wide. Calculate the pressure to which this volume of air must be compressed in order to fit into the air tank. What's your answer in atmospheres? Round your answer to 3 significant digits. The biologist estimates she will need 2700 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. What's your answer in atmospheres? Round your answer to 3 significant digits.

Solution Steps

First, we need to calculate the volume of the spherical air tank. The formula for the volume \( V \) of a sphere is:

\[ V = \frac{4}{3} \pi r^3 \]

Given the diameter of the sphere is 70.0 cm, the radius \( r \) is:

\[ r = \frac{70.0 \, \text{cm}}{2} = 35.0 \, \text{cm} \]

Converting the radius to meters:

\[ r = 0.35 \, \text{m} \]

Now, calculate the volume:

\[ V = \frac{4}{3} \pi (0.35)^3 \]

\[ V \approx \frac{4}{3} \pi (0.042875) \]

\[ V \approx 0.1796 \, \text{m}^3 \]

Since 1 m³ = 1000 L:

\[ V \approx 0.1796 \times 1000 \, \text{L} \]

\[ V \approx 179.6 \, \text{L} \]

Boyle's Law states that \( P_1 V_1 = P_2 V_2 \), where \( P \) is pressure and \( V \) is volume. We need to find the pressure \( P_2 \) to compress 2700 L of air into the 179.6 L tank.

Given:

• \( V_1 = 2700 \, \text{L} \)
• \( V_2 = 179.6 \, \text{L} \)
• \( P_1 = 1 \, \text{atm} \) (assuming initial pressure is 1 atmosphere)

Using Boyle's Law:

\[ P_1 V_1 = P_2 V_2 \]

\[ 1 \times 2700 = P_2 \times 179.6 \]

Solving for \( P_2 \):

\[ P_2 = \frac{2700}{179.6} \]

\[ P_2 \approx 15.03 \, \text{atm} \]

Final Answer