Answer to Question #244249 in Physics for Josh

Question

A typical scuba tank, when fully charged, contains 12.0L of air at 204atm. Assume an empty tank contains air at 34atm and is connected to an air compressor at sea level. The air compressor intakes air from the atmosphere, compresses it to high pressure, and then inputs this high-pressure air into the scuba tank.

Assume the tank remains at the same temperature as the surrounding air during the filling process. If the average flow rate of air from atmosphere into the intake port of the air compressor

is 290L/min, how long will it take to fully charge the scuba tank?

Answer 1

Luckily, we were given the average flow rate, so, we do not need to take into consideration the change in flow rates as the pressure increases.

For 12 L and 204 atm apply the ideal gas law:

$$p_1V=n_1RT,\\\space\\ n_1=\frac{P_1V}{RT}.$$

For 12 L and 34 atm apply the ideal gas law:

$$p_2V=n_2RT,\\\space\\ n_2=\frac{P_2V}{RT}.$$

The amount of substance we need to compress and fit in the tank is

$$n=n_1-n_2=\frac{V}{RT}\bigg(P_1-P_2\bigg)$$

This amount of substance corresponds at normal pressure P takes volume V:

$$PV_a=nRT=\frac{V}{RT}\bigg(P_1-P_2\bigg)·RT=\\\space\\=V(P_1-P_2).$$

The volume at normal pressure (1 atm) is

$$V_a=\frac{V(P_1-P_2)}{P}=\frac{12(204-34)}{1}=2040\text{ L}.$$

The time is

$$T=\frac {V_a}{r}=\frac{2040}{290}=7\text{ min}.$$