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Question Number 29490 by Victor31926 last updated on 09/Feb/18

A gas expands according to the law PV=K (constant). Initialy, v=1000cubic metres and p=40N/m^2 . If the pressure is decreased at the rate of 5N/m^2 /min. find the rate at which the gas is expanding when its volume is 2000cubic metres.

$$\mathrm{A}\:\mathrm{gas}\:\mathrm{expands}\:\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{law}\:\mathrm{PV}=\mathrm{K}\:\left(\mathrm{constant}\right). \\ $$$$\mathrm{Initialy},\:\boldsymbol{\mathrm{v}}=\mathrm{1000cubic}\:\mathrm{metres}\:\mathrm{and}\:\boldsymbol{\mathrm{p}}=\mathrm{40N}/\mathrm{m}^{\mathrm{2}} . \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{pressure}\:\mathrm{is}\:\mathrm{decreased}\:\mathrm{at}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{5N}/\mathrm{m}^{\mathrm{2}} /\mathrm{min}.\:\mathrm{find}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{at}\:\mathrm{which} \\ $$$$\mathrm{the}\:\mathrm{gas}\:\mathrm{is}\:\mathrm{expanding}\:\mathrm{when}\:\mathrm{its}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{2000cubic}\:\mathrm{metres}. \\ $$$$ \\ $$

Answered by Rasheed.Sindhi last updated on 09/Feb/18

PV=K When V=1000 m^3 and P=40N/m^2 K=40 N/m^2 ×1000 m^3 =40000 Nm When V=2000 m^3 P=(K/V)=((40000Nm)/(2000m^3 ))=20N/m^2 Per minute change in P,when V=2000m^3 P changes from 20N/m^2 to 15N/m^2 Changed volume=(K/(Changed pressure)) =((40000Nm)/(15N/m^2 )) =2666(2/3)m^3 Change in volume (in one minute) =2666(2/3)m^3 −2000m^3 =666(2/3)m^3 /min (expanding) Please confirm the answer.

$$\mathrm{PV}=\mathrm{K} \\ $$$$\mathrm{When}\:\mathrm{V}=\mathrm{1000}\:\mathrm{m}^{\mathrm{3}} \:\mathrm{and}\:\mathrm{P}=\mathrm{40N}/\mathrm{m}^{\mathrm{2}} \\ $$$$\mathrm{K}=\mathrm{40}\:\mathrm{N}/\mathrm{m}^{\mathrm{2}} ×\mathrm{1000}\:\mathrm{m}^{\mathrm{3}} =\mathrm{40000}\:\mathrm{Nm} \\ $$$$\mathrm{When}\:\mathrm{V}=\mathrm{2000}\:\mathrm{m}^{\mathrm{3}} \\ $$$$\mathrm{P}=\frac{\mathrm{K}}{\mathrm{V}}=\frac{\mathrm{40000Nm}}{\mathrm{2000m}^{\mathrm{3}} }=\mathrm{20N}/\mathrm{m}^{\mathrm{2}} \\ $$$$\mathrm{Per}\:\mathrm{minute}\:\mathrm{change}\:\mathrm{in}\:\mathrm{P},\mathrm{when}\:\mathrm{V}=\mathrm{2000m}^{\mathrm{3}} \\ $$$$\mathrm{P}\:\mathrm{changes}\:\mathrm{from}\:\mathrm{20N}/\mathrm{m}^{\mathrm{2}} \:\mathrm{to}\:\mathrm{15N}/\mathrm{m}^{\mathrm{2}} \\ $$$$\mathrm{Changed}\:\mathrm{volume}=\frac{\mathrm{K}}{\mathrm{Changed}\:\mathrm{pressure}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{40000Nm}}{\mathrm{15N}/\mathrm{m}^{\mathrm{2}} }\:=\mathrm{2666}\frac{\mathrm{2}}{\mathrm{3}}\mathrm{m}^{\mathrm{3}} \:\:\:\:\: \\ $$$$\mathrm{Change}\:\mathrm{in}\:\mathrm{volume}\:\left(\mathrm{in}\:\mathrm{one}\:\mathrm{minute}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2666}\frac{\mathrm{2}}{\mathrm{3}}\mathrm{m}^{\mathrm{3}} −\mathrm{2000m}^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{666}\frac{\mathrm{2}}{\mathrm{3}}\mathrm{m}^{\mathrm{3}} /\mathrm{min}\:\left(\mathrm{expanding}\right) \\ $$$$\mathrm{Please}\:\mathrm{confirm}\:\mathrm{the}\:\mathrm{answer}. \\ $$

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