Ygtjki

Why was the small council so happy for Tyrion to become the Master of Coin?

Mage Armor with Defense fighting style (for Adventurers League bladeslinger)

Why doesn't Newton's third law mean a person bounces back to where they started when they hit the ground?

Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)

Problem of parity - Can we draw a closed path made up of 20 line segments...

What do you call a Matrix-like slowdown and camera movement effect?

Why did the Germans forbid the possession of pet pigeons in Rostov-on-Don in 1941?

How to find program name(s) of an installed package?

Can divisibility rules for digits be generalized to sum of digits

Is it possible to do 50 km distance without any previous training?

How can bays and straits be determined in a procedurally generated map?

How is the claim "I am in New York only if I am in America" the same as "If I am in New York, then I am in America?

Is it unprofessional to ask if a job posting on GlassDoor is real?

How does one intimidate enemies without having the capacity for violence?

Fencing style for blades that can attack from a distance

In Japanese, what’s the difference between “Tonari ni” (となりに) and “Tsugi” (つぎ)? When would you use one over the other?

Show that if two triangles built on parallel lines, with equal bases have the same perimeter only if they are congruent.

Approximately how much travel time was saved by the opening of the Suez Canal in 1869?

Why Is Death Allowed In the Matrix?

How do we improve the relationship with a client software team that performs poorly and is becoming less collaborative?

How old can references or sources in a thesis be?

Why does Kotter return in Welcome Back Kotter?

Are the number of citations and number of published articles the most important criteria for a tenure promotion?

Why can't I see bouncing of a switch on an oscilloscope?

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

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?

thermodynamics definition volume scaling

share|cite|improve this question

edited 2 days ago

David Z♦

63.9k23136252

asked 2 days ago

paokara moupaokara mou

1436

$endgroup$

add a comment | 

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?

thermodynamics definition volume scaling

share|cite|improve this question

edited 2 days ago

David Z♦

63.9k23136252

asked 2 days ago

paokara moupaokara mou

1436

$endgroup$

add a comment | 

$begingroup$

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?

thermodynamics definition volume scaling

share|cite|improve this question

edited 2 days ago

David Z♦

63.9k23136252

asked 2 days ago

paokara moupaokara mou

1436

$endgroup$

share|cite|improve this question

edited 2 days ago

David Z♦

63.9k23136252

asked 2 days ago

paokara moupaokara mou

1436

share|cite|improve this question

edited 2 days ago

David Z♦

63.9k23136252

asked 2 days ago

paokara moupaokara mou

1436

edited 2 days ago

David Z♦

63.9k23136252

edited 2 days ago

David Z♦

63.9k23136252

asked 2 days ago

paokara moupaokara mou

1436

asked 2 days ago

paokara moupaokara mou

1436

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

edited 2 days ago

answered 2 days ago

lmrlmr

1,074520

$endgroup$

• 
  
  
  

  
  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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
  $endgroup$
  – lmr
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  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
  2 days ago
  
  

  
  
  
  

 | 
show 2 more comments

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "151"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f470452%2fwhy-is-the-ratio-of-two-extensive-quantities-always-intensive%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

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

edited 2 days ago

answered 2 days ago

lmrlmr

1,074520

$endgroup$

• 
  
  
  

  
  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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
  $endgroup$
  – lmr
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  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
  2 days ago
  
  

  
  
  
  

 | 
show 2 more comments

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

edited 2 days ago

answered 2 days ago

lmrlmr

1,074520

$endgroup$

• 
  
  
  

  
  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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yeah, I did not point this out explicitly. I added a few sentences to include the linearity.
  $endgroup$
  – lmr
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @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
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  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
  2 days ago
  
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

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

edited 2 days ago

answered 2 days ago

lmrlmr

1,074520

$endgroup$

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

lmrlmr

1,074520

answered 2 days ago

lmrlmr

1,074520

answered 2 days ago

lmrlmr

1,074520