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Is it necessary for scientific equations to be dimensionally consistent?


studiot

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I am not sure where to post this so have used Philosophy (of Science) to allow latitude in exploring this subject.

I hope it will make a welcome change from the current Philosophy subject we have surely now done to surely death.

Dimensional analysis is a very powerful technique in Science and is one of the things that distinguishes Science from Pure Mathematics, but to repeat the title,

Is it necessary for all equations in Science to be dimensionally consistent?

Edited by studiot
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17 minutes ago, studiot said:

I am not sure where to post this so have used Philosophy (of Science) to allow latitude in exploring this subject.

I hope it will make a welcome change from the current Philosophy subject we have now done to surely death.

Dimensional analysis is a very powerful technique in Science and is one of the things that distinguishes Science from Pure Mathematics, but to repeat the title,

Is it necessary for all equations in Science to be dimensionally consistent?

I’m not sure I understand...if we start comparing quantities of different dimensions like say time and mass or length and velocity we end up straight in cuckoo land like a lot of the crackpots posting in the relativity threads. So in that sense, science has to be consistent dimensionally... but I’m a rookie so I might not be getting the nuances of the question (not like that other thread though ;))

Edited by koti
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I would have to say yes.
Although sometimes, there is a 'cheat' involved which uses the constant of proportionality to make the dimensional analysis equivalent.

As an example, does dimensional analysis determine the units of G, the gravitational constant, or, are the units, m^3*kg^-1*s^-2 , somehow measured, thereby verifying the dimensional analysis of Newton's Gravitational equation ?

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24 minutes ago, koti said:

I’m not sure I understand...if we start comparing quantities of different dimensions like say time and mass or length and velocity we end up straight in cuckoo land like a lot of the crackpots posting in the relativity threads. So in that sense, science has to be consistent dimensionally... but I’m a rookie so I might not be getting the nuances of the question (not like that other thread though ;))

Thanks for the response, koti..  Comparing means creating some sort of  ratio

For instance the statement this glass holds three times as much whisky as that one obviously compares two volumes.

For that purpose I suppose dimensional consistency is required.

 

3 minutes ago, MigL said:

I would have to say yes.
Although sometimes, there is a 'cheat' involved which uses the constant of proportionality to make the dimensional analysis equivalent.

As an example, does dimensional analysis determine the units of G, the gravitational constant, or, are the units, m^3*kg^-1*s^-2 , somehow measured, thereby verifying the dimensional analysis of Newton's Gravitational equation ?

Not sure I'm with you, Mig.

Dimensional analysis is unit free insofar as you can measure in any quantity you like, for instance cords, so long as the measure is the same for both.

 

 

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42 minutes ago, studiot said:

Is it necessary for all equations in Science to be dimensionally consistent?

I don't think this is limited to science. Any equation must be dimensionally correct. You can't say that the area of a carpet is 17 volts x 148ºC 

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1 minute ago, Strange said:

I don't think this is limited to science. Any equation must be dimensionally correct. You can't say that the area of a carpet is 17 volts x 148ºC 

I didn't say you could but not all Scientific equations are, in fact, dimensionally consistent.

 

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The only dimensional analysis I've ever done consists of verifying an equation such that units on either side of the equal sign are equivalent.

That is my understanding of dimensional analysis.
If there is another, proper meaning, as you understand it, please elaborate.
( so we don't go on for several pages as to whether its 'visible' or not )

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7 minutes ago, studiot said:

How about this one

HCL + 10H2O  =  HCL(aq) + 16.61 kcal

The only reason that is not dimensionally consistent (I assume you are referring to the energy term?) is because it is incomplete (one of the things that dimensional analysis can show).

You have only shown the heat released, not the masses (and temperatures) of reagents and products.

I suppose this shows that equations can still be useful, even when incomplete. 

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1 minute ago, Strange said:

The only reason that is not dimensionally consistent (I assume you are referring to the energy term?) is because it is incomplete (one of the things that dimensional analysis can show).

You have only shown the heat released, not the masses (and temperatures) of reagents and products.

I suppose this shows that equations can still be useful, even when incomplete. 

No the equation is complete

Yes the equation shows the masses of both the reagents and products.

What do you think the dimensions of HCL for instance are?

There is no evolved heat on the left hand side so none is shown.

 

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12 minutes ago, studiot said:

How is that?

 

I might be talking crap here because I'm horrible at chemistry but isn't the heat in the water a property therefore the two are consistent dimensionally because they don't have to be compared?
God I think I know where this is going (what MigL mentioned about the other thread) and this time thankfully I will be able to shut up because I suck at chemistry (and math)

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2 minutes ago, studiot said:

Yes the equation shows the masses of both the reagents and products.

No it doesn't. You can't use that to find out how much a hydrogen-chloride molecule weighs.

2 minutes ago, studiot said:

What do you think the dimensions of HCL for instance are?

It is dimensionless. (The number of molecules/atoms.)

3 minutes ago, studiot said:

There is no evolved heat on the left hand side so none is shown.

But there is a mass deficit from the LHS to the RHS.

1 minute ago, koti said:

what MigL mentioned about the other thread

I have no idea what the other thread is, so I don't know where I am going wrong / what I am letting myself in for! :)

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1 minute ago, koti said:

I might be talking crap here because I'm horrible at chemistry but isn't the heat in the water a property therefore the two are consistent dimensionally because they don't have to be compared?
God I think I know where this is going (what MigL mentioned about the other thread) and this time thankfully I will be able to shut because I suck at chemistry (and math)

No the heat evolved is not in the water.

Further you have avoided the question what are the dimensions of HCL etc.

They are not the dimensions of energy.

2 minutes ago, Strange said:

But there is a mass deficit from the LHS to the RHS

So what?

Dimensional analysis requires that the same dimensions appear on both sides of the equation.

It does not require that their values be equal.

 

 

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13 minutes ago, Strange said:

 

I have no idea what the other thread is, so I don't know where I am going wrong / what I am letting myself in for! :)

Light: visible or invisible?

12 minutes ago, studiot said:

No the heat evolved is not in the water.

Further you have avoided the question what are the dimensions of HCL etc.

They are not the dimensions of energy.

I presume the heat evolved is not in the water (in that equation) but I have no idea how to answer what are the "dimensions" of HCL in this scenario, I don;t understand the question. I'll stick to reading in this thread.

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That equation is really four equations in one, a bit like how a complex equation is two in one (real and imaginary).

In this case the four equations each are consistent in their own dimension:

- number of H atoms

- number of O atoms

- number of Cl atoms

- energy

 

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22 minutes ago, Bender said:

That equation is really four equations in one, a bit like how a complex equation is two in one (real and imaginary).

In this case the four equations each are consistent in their own dimension:

- number of H atoms

- number of O atoms

- number of Cl atoms

- energy

 

So where are the units of energy on the left hand side?

 

This equation has no force on the right hand side, but is dimensionally consistent 

 

Force  = rate of change of momentum

 

 

 

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2 hours ago, studiot said:

Is it necessary for all equations in Science to be dimensionally consistent?

Yes.

1 hour ago, studiot said:

HCL + 10H2O  =  HCL(aq) + 16.61 kcal

That's more a matter of sloppy use of language than an equation.

(Lower case "l" in HCl BTW)

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That 'equation' you posted Studiot is only true for one specific circumstance. It is not necessarily true for any temperature or pressure, and therefore incomplete.
It is equivalent to saying New York= Niagara Falls + 8 hrs. Which is only true if by car. It is +45 mi if by plane, or +3 days if walking.
I.E. a totally useless statement, unless other parameters are specified.

Dimensional analysis as I understand it, is a tool, used to make sure you haven't slipped up and made two unequal things equivalent.
The Force=rate of change of momentum, can be simplified to Force=mass*acceleration, which can have both sides in equivalent units.
And that is true always.
I.E.that is effectively, a 'check' of the relationship.

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4 minutes ago, studiot said:

So what?

Dimensional analysis requires that the same dimensions appear on both sides of the equation.

It does not require that their values be equal.

No, but it requires that they appear on both sides. You have mass-energy on the RHS but not the left.

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44 minutes ago, studiot said:

So where are the units of energy on the left hand side?

 

This equation has no force on the right hand side, but is dimensionally consistent 

 

Force  = rate of change of momentum

 

 

 

Your term HCl is actually: 1 H + 1 Cl + 0 O + (energy of one molecule of HCl; for specific circumstances)

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1 hour ago, John Cuthber said:

That's more a matter of sloppy use of language than an equation.

(Lower case "l" in HCl BTW)

Perhaps I'll change my answer to this.

It isn't an actual equation, because it doesn't behave like one. You can't use it to substitute H2O in another equation eg. That would make no sense at all.

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