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# Inverse proportions

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P is a fixed mass of gas at constant temperature is inversely proportional to its volume, V cubed. When v=2, p=125.

A) Find the value of V when P=325

B) Find the value of P when V=10

Aug 21, 2017
edited by Guest  Aug 21, 2017

#1
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From the question we can deduct that the temperature is proportional to the volume of the gas, set:

$$P=\frac{C}{V}$$

With P being pressure, V being the volume, and C being the constant of the gas.

A. Plug $$V=2$$ and $$P=125$$ into the equation:

$$125=\frac{C}{2}$$

$$C=250$$

Therefore the relationship between $$P$$ and $$V$$ is:

$$P=\frac{250}{V}$$

Plug $$P=325$$ into the equation:

$$325=\frac{250}{V}$$

Multiply both sides of the equation by a factor of $$V$$:

$$325V=250$$

Divide both sides by $$325$$:

$$V=\frac{250}{325}=\frac{10}{13}$$ (Unit Volume)

Done :D

B.

Plug $$V=10$$ into the equation:

$$P=\frac{250}{10}=25$$ (Unit Pressure)

Done :D x2

Aug 21, 2017
edited by Jeffes02  Aug 21, 2017
edited by Jeffes02  Aug 21, 2017
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Thanks! It's great that you wrote out every single step which made it easy to follow

Guest Aug 21, 2017
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Haha no thanks :D

Jeffes02  Aug 21, 2017
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The question does not make sense to me. How can the mass of an object be inversely proportional to its volume, much less its volume cubed!

Then again the question calls it a fixed mass, if it is a fixed mass then at a set temperature it would have a fixed voume. There are no variables, everything is fixed.

Overlooking all these 'problems'

$$P=\frac{k}{V^3}\\ 125=\frac{k}{8}\\ k=1000\\ P=\frac{1000}{V^3}\\ A) \\When\;\; P=325\\ 325V^3=1000\\ V=\frac{10}{\sqrt[3]{325}}\\ V\approx 1.454\\~\\ B)\\When\;\;V=10\\ P=\frac{1000}{20^3}=1$$

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Aug 21, 2017
edited by Melody  Aug 21, 2017
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Oops I forgot to put in that P was Pascals (pressure), I hope that makes more sense now

Guest Aug 21, 2017
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But the object in this question is in the state of a gas, neither liquid or solid, therefore shouldn't it be compressible and thus making it possible to have different volume with the same mass?

Jeffes02  Aug 21, 2017
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Thanks guest,

That makes more sense.

The mass of the gas is fixed.

The more pressure on the gas, the less space it takes up. Yes, that works for me. :)

Melody  Aug 21, 2017
edited by Melody  Aug 21, 2017
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Use Boyle's Law

For a fixed mass of gas at constant temperature, the volume is inversely proportional to the pressure. (The written form introduces ambiguities. Volume is measured in cubic units. It is not the cube of the volume.)

Boyle's Law expressed  mathematically:

$$P_1 V_1 = P_2 V_2\\ \text{ }\\ \text{1)}\\ V_2 = \dfrac {P_1 \cdot V_1}{P_2}\\ V_2 = \dfrac {125\; units \cdot 2\;units^3}{325\; units} = 0.76923\; units^3\\$$

$$\text{2)}\\ P_2 = \dfrac {P_1 \cdot V_1}{V_2}\\ P_2 = \dfrac {125\; units \cdot 2\;units^3}{10\; units} = 25\; mass \;units\; per \;area \;units^2 \\$$

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Aug 21, 2017