Your father and your younger brother are confronted with the...

Your father and your younger brother are confronted with the same puzzle. Your father’s garden sprayer and your brother’s water cannon both have tanks with a capacity of 5.00 L (Fig. P19.22). Your father puts a negligible amount of concentrated fertilizer into his tank. They both pour in 4.00 L of water and seal up their tanks, so the tanks also contain air at atmospheric pressure. Next, each uses a hand-operated pump to inject more air until the absolute pressure in the tank reaches 2.40 atm. Now each uses his device to spray out water—not air—until the stream becomes feeble, which it does when the pressure in the tank reaches 1.20 atm. To accomplish spraying out all the water, each finds he must pump up the tank three times. Here is the puzzle: most of the water sprays out after the second pumping. The first and the third pumping-up processes seem just as difficult as the second but result in a much smaller amount of water coming out. Account for this phenomenon.

Key Concepts

Sequential Depressurization Process

The phenomenon described in the puzzle arises because the tank’s contents gradually change composition from mostly water to mostly air after each cycle. In the middle cycle, the ratio of water to air is optimal for converting the extra work done into a large amount of expelled water, whereas in the first and third cycles the same pressure increase yields relatively little water outflow due to the different proportions of air and water remaining in the tank.

Energy Storage in Compressed Air

The energy stored in the compressed air is a function of both the pressure and the volume it occupies. Although each pumping cycle requires a similar amount of work input (raising the pressure by the same increment), the available energy to do work on expelling water varies because the effective gas volume in the tank changes as water is expelled. This results in one particular cycle releasing much more water than the others.

Pressure-Volume Work

When the gas is compressed by pumping air into the tank, work is done on the gas, and energy is stored. As the pressure is subsequently reduced by spraying the water, the gas expands and does work by pushing the water out. The amount of work available for water expulsion during each cycle depends on the pressure–volume change, which is not linearly related to the amount of water present, leading to differing water output in each cycle.

Boyle's Law

This law governs the inverse relationship between the pressure and volume of a gas when the temperature is held constant. In this problem, the air inside the tank is compressed by increasing its pressure, and when the tank is opened to spray water, the expanding air pushes the water out. The pressure changes in the gas, and hence its volume, explain the changing force available to expel water during each cycle.

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