# B How do temperatures add?

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1. Apr 15, 2018

### Terry Bing

I was going through vector and scalar quantities (the way they are taught in high school), and this is how I think students are supposed to understand it:
Scalar quantities are quantities that add like numbers. For e.g. Mass. If I add 100 g of water to a bucket and then add a further 100 g, I could only get 200 g of water as a resultant.
Vector quantities are quantities with magnitude and direction that add according to the triangle rule. If a person travels 5 m in one direction and another 5 m in some other direction, the resultant displacement need not be 10 m and depends on the direction of the two original displacements. The resultant is given by triangle rule.
At this point, one may ask, how is temperature a scalar? Granted, it doesn't have a direction. But how do they add like numbers? If I take some substance at 300 K and mix it with something at 300 K, the resultant temperature is never 600 K.
Can someone comment on how to answer such a question?
I would argue by the following example: consider a person somewhere in Africa travels 5 m north. Another person in the US travels 5 m East. Now I could treat these two displacements as vectors and add them up to get 5√2 m North-East. But this resultant vector doesn't stand for anything physical.
So I guess the question can be asked: what does 'addition of temperatures' mean physically?
Also, are my descriptions of vector and scalar quantities good enough?

Last edited: Apr 15, 2018
2. Apr 15, 2018

### Staff: Mentor

It doesn't mean anything. Temperatures can't be added like masses.

3. Apr 15, 2018

### Terry Bing

So the only argument that can be given to argue that temperature is a scalar is to say that it doesn't have a direction?

4. Apr 15, 2018

### Staff: Mentor

Like a scalar. If you raise the temperature 2 degrees, then 3 degrees more, the total increase is 5 degrees.

If you mix stuff, the result is the average temperature, not the sum. Averages are weighted according to the masses and heat capacities of the substances. See https://en.wikipedia.org/wiki/Heat_capacity

5. Apr 15, 2018

### Drakkith

Staff Emeritus
Nothing. You cannot simply add temperatures together like you can with distance or mass. Temperature is an example of an intensive property of a substance. An intensive property is one that does not depend on the size of the object or the amount of material in it. Contrast this with extensive properties like mass, which directly add or subtract. Cutting an iron bar in half gives you half the mass for each part, but the temperature of each half does not change after the cut.

https://en.wikipedia.org/wiki/Intensive_and_extensive_properties

6. Apr 15, 2018

### Terry Bing

So it is in terms change in temperature, that we can talk of addition : T+ΔT, where T=300 K and ΔT= 30 K would only give 330K ,and nothing else. That is good. Thanks.
There is a small issue of decrease in temperature though. Distances, which are scalars, only increase and never decrease during an objects motion. So Δs would always be +ve. But ΔT can be -ve.

Last edited: Apr 15, 2018
7. Apr 15, 2018

### Terry Bing

I understand. I know about heat capacity, but this is a question asked by students when discussing vectors and scalars. And I gave them the example of person in Africa and U.S. displacement that I mentioned in the original post, to show that one could blindly add two quantities as vectors or scalars, but it is important to know if that addition means anything physically.
SO temperatures don't simply add up when we mix things. But I also needed an example that shows how temperature is a scalar, and in what sense it adds like scalars.

Last edited: Apr 15, 2018
8. Apr 15, 2018

### sophiecentaur

I think it is not so much a 'difference between vectors and scalars' , which is more or less a matter of definition. as a discussion of the differences between various non-vector quantities and how we consider them. There is only a one dimensional scale for scalars but the difference between, say mass and temperature, is only there because of practicalities of how we use them. Calculations with either quantity or any other non vector quantity (and I can't think of an exception) can involve a positive or negative difference. All the non-vector quantities can increase or decrease.
In Maths, the idea of a Number Line has been used in education and numbers on a line can hardly be any different from the Vectors that represent Complex numbers.
Is it really just about the distinction between (Vector and Scalar Quantities ) and (Vectors and Scalars in Maths). I had to use the mathematical parentheses in order to express what I mean.

9. Apr 15, 2018

### Staff: Mentor

You should appeal to the student's common sense. Mix hot water with cold water. You don't get hotter or colder water; you get lukewarm water. Every student knows that. So based on that example, the students themselves can say that the temperatures don't add.

It is a good science lesson for students. Whenever your scientific calculation disagrees with what you experience in daily life, that is a clue that you're applying the science incorrectly. Teach them that and teach them that the wrong answer is to presume that the science is incorrect in such cases.

Also note what @Drakkith said about intensive versus extensive properties. That illustrates that vector versus scalar is true but it is insufficient to describe all the properties. Students should be able to see from ordinary life that intensive/extensive vector/scalar are not the same thing, and that they can exist independently.

10. Apr 15, 2018

### Terry Bing

I am trying to describe vector and scalar physical quantities, without defining vectors and scalars mathematically.
I think I understand. We are mapping on physical quantities to the abstract concept of vectors. So we can perform whatever operations are defined on vector spaces on these physical quantities. However, while making this mapping, we have to careful as to what these operations mean for the physical quantities. And this is a matter of definition.
Hence my question: Since Temperature is defined as a scalar quantity, we can add them (I am not saying we should). But does this addition mean anything physically? Is the addition operation, by definition, mapped on to any physical situation?
@anorlunda says, it is defined in terms of change in temperature of a given system, which seems to make sense to me. In case of vectors, if the velocity of an object changes by Δv , even if the magnitude of Δv remains the same, the final velocity could be different.
You lost me there.

Last edited: Apr 15, 2018
11. Apr 15, 2018

### Terry Bing

The students actually did know through intuition that temperature, upon mixing, should NOT just add up. Their question was, since we know that they just don't add up like masses do, how can we still call them scalars.

12. Apr 15, 2018

### Staff: Mentor

Wonderful teaching opportunity. If I am the student and I'm told vectors do this and scalars do that, and I have an example neither this or that; what can I say?

13. Apr 15, 2018

### Terry Bing

Well, I did make an attempt to answer the question: The resultant temperature when you mix two substance is not what 'adding temperatures' means.
I thought the above example must be convincing.
Moreover, the description I gave of vectors and scalars are not my own, but from the prescribed textbook.
So I had to ask myself. How to explain to a high school student why temperature is a scalar, without using complicated math definitions, and at the same time, without being ambiguous or oversimplifying.

Last edited: Apr 15, 2018
14. Apr 15, 2018

### Staff: Mentor

Isn't the point more that the statement "scalars add" is only true sometimes?
Or that vector versus scalar is an incomplete description of the world?

That is a misapplication. If you use it as an example, it is you who misunderstand vectors and scalars and how they are used.

15. Apr 15, 2018

### DrGreg

Exactly. If you have a car travelling 50 mph north and another travelling 30 mph east, adding their scalar speeds 50+30=80 doesn't make any sense.

16. Apr 15, 2018

### Drakkith

Staff Emeritus
It's a scalar because you can only add or subtract from the temperature. You can't go "5 Kelvin to the left" like you can with a vector. It's really that simple. You don't need any complicated math proofs or anything like that. Remember that scalars and vectors are mathematical concepts and trying to separate them from math can't be done. However, the rules for applying the math to real life scenarios are not the same as the mathematical rules themselves, hence why you can't just add any random physical quantities together and get a correct, useful, and meaningful result. Learning how to correctly apply math to the real world is exactly what learning physics is.

17. Apr 15, 2018

### Terry Bing

I agree that it is indeed a misapplication. I used it as an example of a misapplication. I was trying to show that even though velocity is a vector, adding velocities of two different objects does not make sense, even though you can in principle add two vectors. Please note the sentence, 'But this resultant vector does not stand for anything physical'. Did you think I was giving this as an example of correct vector addition?
If not, could you please elaborate. I will be very grateful.

This is similar to the example I gave.

18. Apr 15, 2018

### Terry Bing

Ok. Thanks.

19. Apr 15, 2018

### sophiecentaur

There are many examples where Vectors are used in Maths and they don't involve anything as prosaic as a quantity and a direction. No one automatically demands a physical interpretation.
Even a simple case of solving the heat flow around a house, there will be positive and negative temperature differences and temperature gradients. My point is that there is a false dichotomy between this division between non-vector quantities. Whether or not you can 'add and subtract' quantities depends totally on the context. Let's face it, Maths is a complete abstraction and we pick and choose when to accept or no the results of the processes we use.

20. Apr 15, 2018

### Terry Bing

Thank you. That makes complete sense. Having a magnitude and direction is an intuitive way of thinking about certain vector quantities we encounter in high school physics, but is not the definition of a vector. And it isn't even the best way to describe vectors even in physics. How would you interpret the 'direction' of the state of a quantum mechanical system, which also behaves as a vector in Hilbert space? Maybe you could imagine its direction if the eigenspace is 3 dimensional or less . But in that case (and in the case of other function spaces), the mathematical definition of vector spaces makes more sense than "something having magnitude and direction".
But then again, there is no point in confusing a high school student with such details.
If you are teaching a kindergarten student about basic geometric shapes, you wouldn't start with Euclid's axiom's.
Yes. Thank you.
The reason the textbook chose to mention 'the way vectors and scalars add' as a part of their definition is because (to paraphrase the author) in circuit diagrams, currents are depicted as having both a magnitude and direction, but they don't add like vectors.

Last edited: Apr 15, 2018