A gas is in a container of volume V0 at pressure P0. It is being pumped out of the container by a piston pump. Each stroke of the piston removes a volume Vs through valve A and then pushes the air out through valve B as shown in Figure P19.74. Derive an expression that relates the pressure Pn of the remaining gas to the number of strokes n that have been applied to the container.
FIGURE P19.74
The expression that relates the pressure
Answer to Problem 74PQ
The expression that relates the pressure
Explanation of Solution
In each stroke, a constant volume of gas is removed from the container of volume
Consider the case when valve A opens and B closes and the gas expands, reaching a new density.
At this stage number of molecules remains constant. After this stage, valve A closes and B opens, and a volume
Write Boyles law.
Here,
Let
Apply Boyle’s law after the first stroke.
Here,
Rearrange equation (I) to get
When the gas is expelled, volume
Apply Boyle’s law for second stroke.
Here,
Rearrange equation (III) to get
Substitute (II) in (IV) to get relation between
Similarly write the expression for pressure of the gas after
In general continuing the pattern, the pressure of the remaining gas will be
Apply Boyle’s law after
Here,
Rearrange above equation to get
Substitute (VI) in (VIII) to get relation between
Conclusion:
Therefore, the expression that relates the pressure
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Chapter 19 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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