Keeping $n$ and $T$ constant, how do I prove that $P prop 1/V$ ? What about keeping $n$ and $P$ constant and showing that $V prop T$ ?

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Keeping $n$ and $T$ constant, how do I prove that $P prop 1/V$ ? What about keeping $n$ and $P$ constant and showing that $V prop T$ ?

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Answer 1 · 2016-10-23T17:44:06

You can check this just from the ideal gas law.

$PV = nRT$

$1)$

If you try to double the pressure, then you are clearly increasing it. As a result, what do you have to do to get $PV = nRT$ back to how it was, assuming $n$ and $T$ remain constant?

$2PV = nRT$

$=> 2P*1/2V = nRT$

Therefore, the volume must have halved to cancel out the pressure doubling. In general, it means $P prop 1/V$ , or that they are inversely proportional.

You can also prove it further:

$P = nRT*1/V$

Therefore, $P prop 1/V$ .

$2)$ Keeping $n$ and $P$ constant, then by now in this answer, you should realize that if $V$ changes, $T$ must change. It can't stay the same.

Can you show that $V prop T$ for constant $P$ and $n$ ? To do that, simply identify the constant in the equation:

$V = (nR)/P*T$

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Keeping $n$ and $T$ constant, how do I prove that $P prop 1/V$ ? What about keeping $n$ and $P$ constant and showing that $V prop T$ ?

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