Question

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`?

Answer 1

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`