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#
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Answer 2

For P prop 1/V, use the ideal gas law: PV = nRT. Rearrange to get P prop 1/V. For V prop T, rearrange the ideal gas law as V prop T when n and P are constant.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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