In a fully surfaced submarine, a certain amount of air at temperature T1 is compressed into a tank of volume V at pressure p1 . As this submarine dives deeply, the temperature of the compressed air drops substantially to T2 , whereas the pressure becomes p2 .

(a) If the submarine needs to surface now, how much water in volume can be blown out of the ballast tanks by compressed air? Assume that the compressed air can be regarded as an ideal gas all the time. [Hint: The key is to find out the volume V2 , of the compressed air when the submarine is in deep water using the equation of state.]

(b) Use the result obtained from the preceding part to determine whether p2 < p1 , p2 = p1 , p2 > p1 . Explain why.

a. V2 = T2/T1 * P1/P2 * V1.

To solve this problem, we will need to use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

a) To find out how much water volume can be blown out of the ballast tanks by the compressed air, we need to find the initial and final volumes of the compressed air.

First, let's assume that the number of moles of air remains constant during the dive. This assumption is reasonable if we neglect any air leakage or gas exchange with the surrounding water.

Applying the ideal gas law to the initial state of the compressed air, we have:
p1 * V = n * R * T1

Similarly, for the final state of the compressed air, we have:
p2 * V2 = n * R * T2

To find V2, we rearrange the equation as follows:
V2 = (n * R * T2) / p2

Now, we need to substitute the values we have for p1, T1, T2, and p2 into these equations. Make sure to convert the temperatures to Kelvin if they are given in Celsius or Fahrenheit.

Once we have calculated V2, we can subtract it from the initial volume V to find the water volume that can be blown out.

b) To determine whether p2 < p1, p2 = p1, or p2 > p1, we compare their values.

If p2 < p1, it means the pressure has decreased as the submarine surfaced. This could happen if the compressed air expanded or if there was an air leak during the dive. In this case, the air cannot blow out any water from the ballast tanks.

If p2 = p1, it means the pressure remains constant during the ascent. This could happen if the compressed air was perfectly sealed and did not expand or contract during the dive. In this case, the air also cannot blow out any water from the ballast tanks.

If p2 > p1, it means the pressure has increased during the ascent. This could happen if the compressed air was further compressed using onboard systems or if the submarine was brought to the surface quickly, trapping the compressed air. In this case, the air can blow out water from the ballast tanks.

By comparing the values of p2 and p1, we can determine which scenario is applicable.