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Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: GeLe5000 on September 25, 2015, 06:01:54 AM

Title: Explanation of differences in heat capacities.
Post by: GeLe5000 on September 25, 2015, 06:01:54 AM
Good morning.

In the book (in French) "Physique, Hecht, de boeck, 2013," there is a table giving the specific heat capacity values (c) of different substances, such as Aluminum (0.90 kJ / kg.K) and Copper ( 0.39 kJ / kg.K).

When they receive the same amount of heat, Aluminum temperature increases less than that of the same quantity of Copper = more heat must be provided to Aluminium in order to get the same increase in temperature.

The molar heat capacity (C) is the specific heat capacity multiplied by the atomic mass. For a solid at room temperature, it is approximately 25 J / kg.K (Dulong and Petit law).

The difference in specific heat capacity between different metals is explained by the fact that heavier atoms are less numerous in the same amount of material. So the heat received is distributed over a smaller number of atoms, each acquiring a greater kinetic energy, responsible for the heating of the thermometer. So Copper (atomic mass: 63.5) will give a higher temperature than Aluminum (atomic mass: 27).

So I do not understand.

If one pierces a hole in a Copper block to place there the sensitive part of a thermometer, and the same hole in an Aluminum block, the thermometer will be exposed to the same metal surface. Using the ratio of the atomic masses (63.5 / 27), the thermometer is exposed to 2.35 times more collisions with the atoms of Aluminum (more atoms) than those of Copper (fewer atoms).
If thermal energy is distributed over a smaller number of atoms in the case of Copper, causing a greater increase in their kinetic energy, this increase is offset by the greater number of atoms present on the same surface of Aluminum.

So the same amount of energy should be transmitted to the thermometer. Right or not ?
Title: Re: Explanation of differences in heat capacities.
Post by: Borek on September 25, 2015, 06:15:03 AM
Sounds like you are confused about extensive and intensive properties.

https://en.wikipedia.org/wiki/Intensive_and_extensive_properties
Title: Re: Explanation of differences in heat capacities.
Post by: GeLe5000 on September 25, 2015, 01:37:05 PM
The temperature is an intensive property. It's the concentration in thermic (or thermal ?) energy. I don't quite see the problem with that… But it's a stimulus, thank you very much.

Anyway, I think that I've found something.

Block "A" and "a" have the same mass and receive the same amount of thermic energy, Q.

The atomic mass of "A" is twice the atomic mass of "a". The specific heat capacity of block "A" will be lower than block "a" specific heat capacity.

Under the same surface S, there are 3 atoms of "A" or 6 atoms of "a".

      S                    S
   --------------------      --------------------
   A      A      A         a  a  a   a  a  a

The thermometer in contact with these surfaces gives TA higher than Ta.

It implies that the energy per atom of "A" (EA) must be strictly higher than twice the energy per atom of "a" (Ea).

Indeed,

EA can't be lower than Ea, or equal to Ea, since a higher mass means higher kinetic energy. The same if EA is between 1 and 2 x Ea.

EA can't be twice Ea since in this case TA would be equal to Ta : If EA = 2 x Ea, the total amount of energy received by the thermometer on block A is 3 x 2 x Ea = 6 x Ea, the same as received on block a. Impossible since TA > Ta.
If EA = 2,000…001 x Ea
3 x 2,000…001 = 6,000…003 > 6 x Ea ---> TA > Ta.
Title: Re: Explanation of differences in heat capacities.
Post by: Borek on September 25, 2015, 04:09:13 PM
Number of atoms in contact doesn't matter. Just because there are more/less atoms doesn't mean they will transfer more/less energy.

If the contact surface would matter, thermometer would give different readings depending on how deeply it is immersed. That's not what we observe.
Title: Re: Explanation of differences in heat capacities.
Post by: GeLe5000 on September 25, 2015, 04:23:54 PM
I'm sorry, Borek, if you plunge the thermometer more deeply, the surface in contact won't change. The sensitive part of the thermometer is always exposed to the same number of atoms or molécules. So, the number of atoms in contact DOES matter.



Title: Re: Explanation of differences in heat capacities.
Post by: mjc123 on September 25, 2015, 06:05:11 PM
Do you think the thermometer gets heat only from the few atoms with which it is in direct contact? What happens to them? Heat flows to them from the atoms in the next layer, and so on. So heat flows from the whole block to the thermometer until it is at the same temperature as the block. (We assume the thermal mass of the thermometer is much less than that of the block, or it wouldn't be a good measurer of temperature.)
Title: Re: Explanation of differences in heat capacities.
Post by: Borek on September 26, 2015, 02:30:18 AM
Contact surface matters only when it comes to the heat transfer speed. Yes, higher contact surface means higher transfer speed, but it doesn't change the final temperature at equilibrium.
Title: Re: Explanation of differences in heat capacities.
Post by: GeLe5000 on September 26, 2015, 09:59:50 AM
Thank you very much for your help.
I still haven't quite understood your point of view, but anyway I've come to something new. I was wrong when I was thinking that Al atoms would be more numerous (because smaller) then Copper atoms because Copper has a higher atomic mass. In fact, it's the opposite, as shown by the densities and atomic volumes. There are 1,41 more Cu atoms in a same volume, and 1,26 more on the same surface in contact with the thermometer (If my calculations are good because I'm not good in mathematics, I'm only good in calculation mistakes).

Thank you again.


Title: Re: Explanation of differences in heat capacities.
Post by: mjc123 on September 27, 2015, 04:02:34 PM
That's interesting but irrelevant, since, for the reasons stated, the temperature doesn't depend on the number of atoms in contact with the thermometer.
If you can be more specific about what you didn't understand in the explanation, perhaps we can try and make it clearer.
Title: Re: Explanation of differences in heat capacities.
Post by: GeLe5000 on September 27, 2015, 04:31:16 PM
For now, I still can't believe that "the temperature doesn't depend on the number of atoms in contact with the thermometer". After one night sleep, I will give you the calculations on the basis of density values and the reasoning relatively to the text in my Physics book (which, perhaps is misleading me).

Have a good night.

Title: Re: Explanation of differences in heat capacities.
Post by: DrCMS on September 27, 2015, 05:35:03 PM
Quote from: GeLe5000 on September 27, 2015, 04:31:16 PM
For now, I still can't believe that "the temperature doesn't depend on the number of atoms in contact with the thermometer".

It does not matter what you believe or what calculation you present.  You have been told the answer and it will still be true even if you do not believe it.
Title: Re: Explanation of differences in heat capacities.
Post by: Enthalpy on September 27, 2015, 07:12:22 PM
Hi GeLE5000, "The number of collisions" seems to be what misleads you.

You can understand temperature as an energy per degree of freedom, since this energy spreads out equally at equilibrium - at least for the degree of freedom not too coarsely quantified, which is the case for atom translations.

Whether you have more or less atoms in contact won't change the temperature.

But more atoms in a chunk of matter offer more degrees of freedom that store more heat for the same temperature variation, hence a bigger capacity.
Title: Re: Explanation of differences in heat capacities.
Post by: GeLe5000 on September 28, 2015, 02:44:35 AM
Despite the discouraging message from DrCMS, I show the reasoning. You can tell me if my understanding of the textbook is wrong.

1)   First, there is this sentence, about a modern interpretation of the observation by Joseph Black that water and Iron had different specific heat capacities : "Heat is spread over the molecules. The more numerous the molecules, the less the increase in thermal energy per molecule, thus less kinetic energy and a lower increase in temperature".
I can't avoid seeing the thermometer surrounded by molecules or atoms, whose number seems important.

2)   "Thus, for the same amount of heat, 1 Kg of atoms of low mass undergoes a temperature increase lower than in the case of 1 Kg of atoms having a higher mass (thus less numerous). The first one has a higher specific heat capacity."
Same remark as above when I read "thus less numerous".

3)   When I read "atoms having a higher mass" I forgot something that I had learned before. A higher mass doesn't mean that the atoms are bigger. On the contrary, the mass is concentrated in the nucleus and the attraction of the electrons reduces the atomic volume. So, I was totally wrong when I supposed that Aluminium atoms were smaller, thus more numerous, and would counteract the higher kinetic energy of Copper atoms, through their higher number.

4)   Moles in 1 Kg.

Moles in 1 Kg of Cu (AM : 63,5) : 1000/63,5 = 15,55 moles
Moles in 1 Kg of Al (AM : 27) : 1000/27 = 37,04 moles

37,04/15,55 = 2,38

So, there are 2,38 times more Al atoms in 1 Kg. Cu atoms are less numerous in 1 Kg, thus more energetic.

5)   But, when considering the density values :

Cu : 8,96 g/ml
Al : 2,7 g/ml

Volume of 1 Kg Cu : 1000/8,96 = 111,61 ml
Volume of 1 Kg Al : 1000/2,7 = 370,37 ml

Cu : 15,55 moles/111,61 ml = 0,139 moles/ml
Al : 37,04 moles/ 370,37 ml = 0,1 moles/ml

Not only Cu atoms are more energetic, they are also more numerous in a volume surrouding the sensor of a thermometer, even if they less numerous in 1 Kg.
Title: Re: Explanation of differences in heat capacities.
Post by: Borek on September 28, 2015, 04:24:58 AM
You are making several mistakes, but it is still not clear to me what they are. I have a feeling one of them is assumption that the temperature change is defined by the ΔT=amount of heat/heat capacity. That's OK if we know the amount of heat transferred, but amount of heat transferred is limited - you can't heat the other object to the temperature higher than the object that is a source of heat. Most of your points ignore that fact, as if the maximum attainable temperature was not limited this way.

You are also wrong about the average kinetic energy of a molecule (or an atom, if we are dealing with an atomic substance). No, it doesn't depend on the atom/molecule mass, it is always [itex]\frac 3 2 kT[/itex], where k is a Boltzman constant and T is the absolute temperature. Note the while the kinetic energy doesn't depend on the atom/molecule identity its velocity does, as the lower the mass, the faster the atom/molecule must be to have the same kinetic energy. Statements like

Quote from: GeLe5000 on September 28, 2015, 02:44:35 AM
Aluminium atoms were smaller, thus more numerous, and would counteract the higher kinetic energy of Copper atoms

are clearly wrong, as the average kinetic energy is exactly the same.
Title: Re: Explanation of differences in heat capacities.
Post by: mjc123 on September 28, 2015, 04:54:24 AM
Quote
1)   First, there is this sentence, about a modern interpretation of the observation by Joseph Black that water and Iron had different specific heat capacities : "Heat is spread over the molecules. The more numerous the molecules, the less the increase in thermal energy per molecule, thus less kinetic energy and a lower increase in temperature".
I can't avoid seeing the thermometer surrounded by molecules or atoms, whose number seems important.
This seems to be your key misunderstanding. Yes, there are more atoms in 1 kg of Al than 1 kg of Cu, so the specific heat capacity of Al is greater. But the number of atoms that matters is all the atoms in the sample. I ask you again, do you really think that only the atoms in immediate contact with the thermometer give up heat to it? If heat can flow from these atoms to the thermometer, why not to these atoms from those in the adjacent layer (and thence to the thermometer), and so on. Heat flows from the sample to the thermometer (or the other way) until they are at the same temperature, and heat flows within the sample so that the whole sample is at the same temperature.
For a meaningful temperature measurement, it is necessary that the heat flow to/from the thermometer does not affect the sample temperature, therefore that the thermal mass of the sample (i.e. mass x specific heat capacity) is much greater than that of the thermometer. Provided this is satisfied, it doesn't matter how many atoms (how big a sample) you have.
The number of atoms in immediate contact with the thermometer will affect how quickly equilibrium is reached, but it will not affect the equilibrium temperature. You seem to be missing the fundamental distinction between equilibrium (thermodynamics) and kinetics (the rates of processes), and the factors that influence each.
Title: Re: Explanation of differences in heat capacities.
Post by: GeLe5000 on September 28, 2015, 09:57:53 AM
Hello. The conclusion, I suppose. I hope that I haven't bothered you too much.
From your last posts I conclude that when reading the rest of my textbook, I'll have to try to understand mechanical statistics (degrees of freedom, 3/2 kT…) and the interplay of thermodynamics and kinetics.

I'd have liked to show you a graph that helps me visualize the fact that temperature is a measure of the concentration of thermic energy, but wasn't able to insert the image. On the same graph, two series of points : the ratios of specific heat capacities for couples of metals (Al, Cu, Au, Fe, Pb, Pt, Ag) and the product (ratio moles in 1 Kg) x (ratio moles in 1 ml). There are 21 couples. As you can see on the Table, except for Lead (the three highest points), there's a good correlation between the two series.

I : ratios of specific heat capacities
II : product (ratio moles in 1 Kg) x (ratio moles in 1 ml)

Couple   I       II
Au/Fe   0,28   0,2
Au/Ag   0,56   0,55
Pb/Ag   0,56   0,29
Pt/Ag   0,61   0,605
Cu/Fe   0,83   0,88
Au/Pt   0,93   0,91
Pb/Pt   0,93   0,47
Au/Pb   1   1,9
Cu/Ag   1,7   2,4
Al/Fe   1,9   1,47
Fe/Ag   2,04   2,7
Al/Cu   2,3   1,67
Cu/Pt   2,8   3,94
Cu/Au   3   4,37
Cu/Pb   3   8,34
Fe/Pt   3,35   4,4
Fe/Pb   3,6   9,4
Al/Ag   3,9   4
Al/Pt   6,4   6,6
Al/Au   6,9   7,29
Al/Pb   6,9   13,9

Title: Re: Explanation of differences in heat capacities.
Post by: mjc123 on September 28, 2015, 11:22:56 AM
Temperature is a measure of the average thermal (kinetic) energy of the molecules, not the concentration of thermal energy (if by that you mean energy per unit volume). Consider a lump of gold at 300K in thermal equilibrium with gaseous helium at 300K. Or He at 300K and 0.1 atm in thermal equilibrium, via a partition, with He at 300K and 10 atm. The average kinetic energy of the atoms is the same (that's the definition of temperature), but the concentration obviously isn't.
If you look at the table in this article https://en.wikipedia.org/wiki/Heat_capacity you will see that for many metals, particularly transition metals, the volumetric heat capacity is roughly similar, at about 2-2.5 J/cm3/K. This is because their molar volumes are roughly similar, around 10 cm3/mol or so, and their molar heat capacities are all around 3R. In that case your plot reduces to a correlation between heat capacity ratio and inverse ratio of atomic weights, as we expect. Molar volumes tend to be higher for the s and p block metals, which is why lead doesn't fit the trend.