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(A) Since the temperature remains constant, we use Boyle's Law, $PV = 	ext{constant}$.Initially, the gas occupies volume $V$ at pressure $P$. When th

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$P\left(\frac{V}{V+v}\right)^n$

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(A) Since the temperature remains constant, we use Boyle's Law, $PV = 	ext{constant}$.Initially, the gas occupies volume $V$ at pressure $P$. When the pump performs one stroke, the gas expands to fill the total volume $(V + v)$.Let $P_1$ be the pressure after the first stroke:$P \cdot V = P_1 \cdot (V + v)$$P_1 = P \left( \frac{V}{V + v} ight)$After the second stroke, the gas at pressure $P_1$ in volume $V$ expands to volume $(V + v)$:$P_1 \cdot V = P_2 \cdot (V + v)$$P_2 = P_1 \left( \frac{V}{V + v} ight) = P \left( \frac{V}{V + v} ight)^2$Following this pattern, after $n$ strokes, the pressure $P_n$ will be:$P_n = P \left( \frac{V}{V + v} ight)^n$

An ideal gas is enclosed in a container of volume $V$ at a pressure $P$. It is being pumped out of the container by using a pump with stroke volume $v$. What is the final pressure in the container after $n$ strokes of the pump? (Assume temperature remains constant.)

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