Why is the ratio of two extensive quantities always intensive?Definition of “intensive” and “extensive” propertiesAre $G$, $F$ and $H$ (thermodynamics potentials) extensive quantities?Is speed an intensive property?Why is density an intensive property?Extensive variables in thermodynamicsPressure: extensive or intensive property?Is heat an extensive or intensive property?Why we can't multiply two extensive quantities together?Is length an extensive property?Rigorous definition of intensive and extensive quantities in thermodynamics

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Why is the ratio of two extensive quantities always intensive?


Definition of “intensive” and “extensive” propertiesAre $G$, $F$ and $H$ (thermodynamics potentials) extensive quantities?Is speed an intensive property?Why is density an intensive property?Extensive variables in thermodynamicsPressure: extensive or intensive property?Is heat an extensive or intensive property?Why we can't multiply two extensive quantities together?Is length an extensive property?Rigorous definition of intensive and extensive quantities in thermodynamics













12












$begingroup$


Is this something that we observe that always happens or is there some fundamental reason for two extensive quantities to give an intensive when divided?










share|cite|improve this question











$endgroup$
















    12












    $begingroup$


    Is this something that we observe that always happens or is there some fundamental reason for two extensive quantities to give an intensive when divided?










    share|cite|improve this question











    $endgroup$














      12












      12








      12


      1



      $begingroup$


      Is this something that we observe that always happens or is there some fundamental reason for two extensive quantities to give an intensive when divided?










      share|cite|improve this question











      $endgroup$




      Is this something that we observe that always happens or is there some fundamental reason for two extensive quantities to give an intensive when divided?







      thermodynamics definition volume scaling






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Apr 4 at 11:04









      David Z

      63.9k23136252




      63.9k23136252










      asked Apr 4 at 6:19









      paokara moupaokara mou

      1587




      1587




















          1 Answer
          1






          active

          oldest

          votes


















          18












          $begingroup$

          It is mainly a mathematical reason. Extensive quantities grow with system size. If two quantities scale in the same way with a variable (in this case system size), it cancels out in the division.



          Mini-example: $A$ and $B$ are extensive physical quantities both dependent on $n$. Their ratio is called $C = A / B$. If you scale the system up, $A$ and $B$ grow by a factor of $n$. What happens to $C$?



          $fracA cdot nB cdot n = fracAB$



          $C$ stays the same, irrespective of $n$. Hence, $C$ is intensive. The most common physical example is mass and volume, which scale with system size and still exhibit the same ratio, the density.



          EDIT including the comment of probably_someone: The argumentation is particularly true since by definition an extensive quantity grows linearly with system size. This justifies the proportionality that I presented in the mini-example.






          share|cite|improve this answer











          $endgroup$








          • 8




            $begingroup$
            In particular, this is true because "extensive" is specifically defined as growing linearly with system size (see e.g. en.wikipedia.org/wiki/Intensive_and_extensive_properties), which raises the question: what do we call a property that grows nonlinearly with system size (for example, as the square of the volume)?
            $endgroup$
            – probably_someone
            Apr 4 at 10:21










          • $begingroup$
            Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
            $endgroup$
            – lmr
            Apr 4 at 10:50










          • $begingroup$
            Technically couldn't the linear relations have different "slopes", so that the part dependant on the size still cancels, but there will be some extra constant factor multiplying your ratio there?
            $endgroup$
            – Aaron Stevens
            Apr 4 at 11:32










          • $begingroup$
            @AaronStevens Well mathematically, it is definitely possible. I can't think of a suitable example right now though. But as you pointed out yourself, the ratio will still remain intensive.
            $endgroup$
            – lmr
            Apr 4 at 12:04






          • 4




            $begingroup$
            @AaronStevens Whatever the factor is is already included in $A$ and $B$ in this answer. And in particular, if the quantities have different units then not only are the slopes different, they have different units so they are clearly very different, but all that is automatically accounted for in the division. Having the same unit but different magnitude is no different.
            $endgroup$
            – JiK
            Apr 4 at 12:51












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          1 Answer
          1






          active

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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

          votes









          18












          $begingroup$

          It is mainly a mathematical reason. Extensive quantities grow with system size. If two quantities scale in the same way with a variable (in this case system size), it cancels out in the division.



          Mini-example: $A$ and $B$ are extensive physical quantities both dependent on $n$. Their ratio is called $C = A / B$. If you scale the system up, $A$ and $B$ grow by a factor of $n$. What happens to $C$?



          $fracA cdot nB cdot n = fracAB$



          $C$ stays the same, irrespective of $n$. Hence, $C$ is intensive. The most common physical example is mass and volume, which scale with system size and still exhibit the same ratio, the density.



          EDIT including the comment of probably_someone: The argumentation is particularly true since by definition an extensive quantity grows linearly with system size. This justifies the proportionality that I presented in the mini-example.






          share|cite|improve this answer











          $endgroup$








          • 8




            $begingroup$
            In particular, this is true because "extensive" is specifically defined as growing linearly with system size (see e.g. en.wikipedia.org/wiki/Intensive_and_extensive_properties), which raises the question: what do we call a property that grows nonlinearly with system size (for example, as the square of the volume)?
            $endgroup$
            – probably_someone
            Apr 4 at 10:21










          • $begingroup$
            Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
            $endgroup$
            – lmr
            Apr 4 at 10:50










          • $begingroup$
            Technically couldn't the linear relations have different "slopes", so that the part dependant on the size still cancels, but there will be some extra constant factor multiplying your ratio there?
            $endgroup$
            – Aaron Stevens
            Apr 4 at 11:32










          • $begingroup$
            @AaronStevens Well mathematically, it is definitely possible. I can't think of a suitable example right now though. But as you pointed out yourself, the ratio will still remain intensive.
            $endgroup$
            – lmr
            Apr 4 at 12:04






          • 4




            $begingroup$
            @AaronStevens Whatever the factor is is already included in $A$ and $B$ in this answer. And in particular, if the quantities have different units then not only are the slopes different, they have different units so they are clearly very different, but all that is automatically accounted for in the division. Having the same unit but different magnitude is no different.
            $endgroup$
            – JiK
            Apr 4 at 12:51
















          18












          $begingroup$

          It is mainly a mathematical reason. Extensive quantities grow with system size. If two quantities scale in the same way with a variable (in this case system size), it cancels out in the division.



          Mini-example: $A$ and $B$ are extensive physical quantities both dependent on $n$. Their ratio is called $C = A / B$. If you scale the system up, $A$ and $B$ grow by a factor of $n$. What happens to $C$?



          $fracA cdot nB cdot n = fracAB$



          $C$ stays the same, irrespective of $n$. Hence, $C$ is intensive. The most common physical example is mass and volume, which scale with system size and still exhibit the same ratio, the density.



          EDIT including the comment of probably_someone: The argumentation is particularly true since by definition an extensive quantity grows linearly with system size. This justifies the proportionality that I presented in the mini-example.






          share|cite|improve this answer











          $endgroup$








          • 8




            $begingroup$
            In particular, this is true because "extensive" is specifically defined as growing linearly with system size (see e.g. en.wikipedia.org/wiki/Intensive_and_extensive_properties), which raises the question: what do we call a property that grows nonlinearly with system size (for example, as the square of the volume)?
            $endgroup$
            – probably_someone
            Apr 4 at 10:21










          • $begingroup$
            Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
            $endgroup$
            – lmr
            Apr 4 at 10:50










          • $begingroup$
            Technically couldn't the linear relations have different "slopes", so that the part dependant on the size still cancels, but there will be some extra constant factor multiplying your ratio there?
            $endgroup$
            – Aaron Stevens
            Apr 4 at 11:32










          • $begingroup$
            @AaronStevens Well mathematically, it is definitely possible. I can't think of a suitable example right now though. But as you pointed out yourself, the ratio will still remain intensive.
            $endgroup$
            – lmr
            Apr 4 at 12:04






          • 4




            $begingroup$
            @AaronStevens Whatever the factor is is already included in $A$ and $B$ in this answer. And in particular, if the quantities have different units then not only are the slopes different, they have different units so they are clearly very different, but all that is automatically accounted for in the division. Having the same unit but different magnitude is no different.
            $endgroup$
            – JiK
            Apr 4 at 12:51














          18












          18








          18





          $begingroup$

          It is mainly a mathematical reason. Extensive quantities grow with system size. If two quantities scale in the same way with a variable (in this case system size), it cancels out in the division.



          Mini-example: $A$ and $B$ are extensive physical quantities both dependent on $n$. Their ratio is called $C = A / B$. If you scale the system up, $A$ and $B$ grow by a factor of $n$. What happens to $C$?



          $fracA cdot nB cdot n = fracAB$



          $C$ stays the same, irrespective of $n$. Hence, $C$ is intensive. The most common physical example is mass and volume, which scale with system size and still exhibit the same ratio, the density.



          EDIT including the comment of probably_someone: The argumentation is particularly true since by definition an extensive quantity grows linearly with system size. This justifies the proportionality that I presented in the mini-example.






          share|cite|improve this answer











          $endgroup$



          It is mainly a mathematical reason. Extensive quantities grow with system size. If two quantities scale in the same way with a variable (in this case system size), it cancels out in the division.



          Mini-example: $A$ and $B$ are extensive physical quantities both dependent on $n$. Their ratio is called $C = A / B$. If you scale the system up, $A$ and $B$ grow by a factor of $n$. What happens to $C$?



          $fracA cdot nB cdot n = fracAB$



          $C$ stays the same, irrespective of $n$. Hence, $C$ is intensive. The most common physical example is mass and volume, which scale with system size and still exhibit the same ratio, the density.



          EDIT including the comment of probably_someone: The argumentation is particularly true since by definition an extensive quantity grows linearly with system size. This justifies the proportionality that I presented in the mini-example.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Apr 4 at 10:49

























          answered Apr 4 at 6:27









          lmrlmr

          1,072520




          1,072520







          • 8




            $begingroup$
            In particular, this is true because "extensive" is specifically defined as growing linearly with system size (see e.g. en.wikipedia.org/wiki/Intensive_and_extensive_properties), which raises the question: what do we call a property that grows nonlinearly with system size (for example, as the square of the volume)?
            $endgroup$
            – probably_someone
            Apr 4 at 10:21










          • $begingroup$
            Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
            $endgroup$
            – lmr
            Apr 4 at 10:50










          • $begingroup$
            Technically couldn't the linear relations have different "slopes", so that the part dependant on the size still cancels, but there will be some extra constant factor multiplying your ratio there?
            $endgroup$
            – Aaron Stevens
            Apr 4 at 11:32










          • $begingroup$
            @AaronStevens Well mathematically, it is definitely possible. I can't think of a suitable example right now though. But as you pointed out yourself, the ratio will still remain intensive.
            $endgroup$
            – lmr
            Apr 4 at 12:04






          • 4




            $begingroup$
            @AaronStevens Whatever the factor is is already included in $A$ and $B$ in this answer. And in particular, if the quantities have different units then not only are the slopes different, they have different units so they are clearly very different, but all that is automatically accounted for in the division. Having the same unit but different magnitude is no different.
            $endgroup$
            – JiK
            Apr 4 at 12:51













          • 8




            $begingroup$
            In particular, this is true because "extensive" is specifically defined as growing linearly with system size (see e.g. en.wikipedia.org/wiki/Intensive_and_extensive_properties), which raises the question: what do we call a property that grows nonlinearly with system size (for example, as the square of the volume)?
            $endgroup$
            – probably_someone
            Apr 4 at 10:21










          • $begingroup$
            Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
            $endgroup$
            – lmr
            Apr 4 at 10:50










          • $begingroup$
            Technically couldn't the linear relations have different "slopes", so that the part dependant on the size still cancels, but there will be some extra constant factor multiplying your ratio there?
            $endgroup$
            – Aaron Stevens
            Apr 4 at 11:32










          • $begingroup$
            @AaronStevens Well mathematically, it is definitely possible. I can't think of a suitable example right now though. But as you pointed out yourself, the ratio will still remain intensive.
            $endgroup$
            – lmr
            Apr 4 at 12:04






          • 4




            $begingroup$
            @AaronStevens Whatever the factor is is already included in $A$ and $B$ in this answer. And in particular, if the quantities have different units then not only are the slopes different, they have different units so they are clearly very different, but all that is automatically accounted for in the division. Having the same unit but different magnitude is no different.
            $endgroup$
            – JiK
            Apr 4 at 12:51








          8




          8




          $begingroup$
          In particular, this is true because "extensive" is specifically defined as growing linearly with system size (see e.g. en.wikipedia.org/wiki/Intensive_and_extensive_properties), which raises the question: what do we call a property that grows nonlinearly with system size (for example, as the square of the volume)?
          $endgroup$
          – probably_someone
          Apr 4 at 10:21




          $begingroup$
          In particular, this is true because "extensive" is specifically defined as growing linearly with system size (see e.g. en.wikipedia.org/wiki/Intensive_and_extensive_properties), which raises the question: what do we call a property that grows nonlinearly with system size (for example, as the square of the volume)?
          $endgroup$
          – probably_someone
          Apr 4 at 10:21












          $begingroup$
          Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
          $endgroup$
          – lmr
          Apr 4 at 10:50




          $begingroup$
          Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
          $endgroup$
          – lmr
          Apr 4 at 10:50












          $begingroup$
          Technically couldn't the linear relations have different "slopes", so that the part dependant on the size still cancels, but there will be some extra constant factor multiplying your ratio there?
          $endgroup$
          – Aaron Stevens
          Apr 4 at 11:32




          $begingroup$
          Technically couldn't the linear relations have different "slopes", so that the part dependant on the size still cancels, but there will be some extra constant factor multiplying your ratio there?
          $endgroup$
          – Aaron Stevens
          Apr 4 at 11:32












          $begingroup$
          @AaronStevens Well mathematically, it is definitely possible. I can't think of a suitable example right now though. But as you pointed out yourself, the ratio will still remain intensive.
          $endgroup$
          – lmr
          Apr 4 at 12:04




          $begingroup$
          @AaronStevens Well mathematically, it is definitely possible. I can't think of a suitable example right now though. But as you pointed out yourself, the ratio will still remain intensive.
          $endgroup$
          – lmr
          Apr 4 at 12:04




          4




          4




          $begingroup$
          @AaronStevens Whatever the factor is is already included in $A$ and $B$ in this answer. And in particular, if the quantities have different units then not only are the slopes different, they have different units so they are clearly very different, but all that is automatically accounted for in the division. Having the same unit but different magnitude is no different.
          $endgroup$
          – JiK
          Apr 4 at 12:51





          $begingroup$
          @AaronStevens Whatever the factor is is already included in $A$ and $B$ in this answer. And in particular, if the quantities have different units then not only are the slopes different, they have different units so they are clearly very different, but all that is automatically accounted for in the division. Having the same unit but different magnitude is no different.
          $endgroup$
          – JiK
          Apr 4 at 12:51


















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