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Problem 53 Medium Difficulty

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure $ P $ of the gas is inversely proportional to the volume $ V $ of the gas.
(a) Suppose that the pressure of a sample of air that occupies $ 0.106 m^3 at 25^{\circ} C $ is 50 kPa. Write $ V $ as a function of $ P. $
(b) Calculate $ dV\mid dP $ when $ P $ = 50 kPA. What is the meaning of the derivative? What are its units?


(a) $$P=\frac{5.3}{V} \text { and } V=\frac{5.3}{P}$$
(b) $$P=50, \frac{d V}{d P}=-\frac{5.3}{50^{2}}=-0.00212$$

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Video Transcript

it's clear. So when you married here. So we have PV equals K. He has a constant. We're gonna substitute 50 for P and 500.106 for ve me get Kate is equal to 5.3 and B is equal to 5.3 Overpay for part B V is equal to 5.3. Overpay p is equal to 50 Killer Paschal's. We're going to find the derivative of RV. This becomes equal to negative 5.3 he the negative to power. This could be rewritten as negative 5.3 over a piece square. Then we plug in 50 when we got negative 0.212 meters cubed over Killa Pasko's. And this represents the rate of change of the volume in respect to the pressure.