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#?

1 Answer
Oct 23, 2016

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#