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I have read that regarding the equation of state $PV=nRT$, since $V$ is an extensive quantity $P$ should be an intensive one because the product of extensive quantities is inherently non-linear.

Could someone explain this statement?

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Assume for now that both P,V are extensive quantities. By the definition of an extensive quantity if the size of system is increased by a factor of λ the extensive quantity is multiplied by λ but the intensive quantities remain the same. So P->(λP) and V->(λV) therefore $$(PV)\rightarrow λ^{2}(PV) $$ This λ square might be the non-linearity that the statement is reffering to.

n->(λn), so if it is also known that T is an intensive quantity then P should be necessarily an intensive quantity

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