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This is probably a stupid question, but why do different materials have different specific heat capacities?

To better understand my question let's say that I have $1$ kg of copper and $1$ kg of water. The amount of heat required to raise the temperature of the water by $1$ degree is about $10.8$ times that of copper (see footnote). Where is this extra energy that's required to increase the water's temperature by one degree compared to the copper "stored"?

Moreover, suppose if I have $1$ kg of ice and $1$ kg of liquid water where both substances are made up of the same molecules. Even though it is the same "substance" they still have different specific heat capacities. Is it simply because the numbers of atoms per kg in the substances are different?

Footnote: $C_{\text{Cu}}=385$ J/(kg K), $C_{\text{H$_2$O}}=4.19\cdot 10^{3}$ J/(kg K).

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5 Answers 5

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Primarily because 1 kg of water has more atoms than 1 kg of copper.

For ordinary solids at or above room temperature, the molar heat capacity is approximately the same (Dulong and Petit's law), three times the gas constant, about 25 J/K. It is because the sum of potential and kinetic energy per atom is $3k_BT$ in the harmonic approximation.

Hydrogen in water or ice is a bit different. It is so light that quantum effects come inte play, equipartition does not apply.

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  • $\begingroup$ Waddup Pieter, when you say 'Primarily because 1 kg of water has more atoms than 1 kg of copper.' is that molar heat capacity or specific heat capacity? $\endgroup$
    – CountDOOKU
    Apr 18, 2019 at 11:53
  • $\begingroup$ @FredWeasley It is my answer on the OP's question: the specific heat capacity per unit weight. And I added that the heat capacity per mole or per atom is approximately the same for solids at or above room temperature. $\endgroup$
    – user137289
    Apr 18, 2019 at 13:34
  • $\begingroup$ so the element's atom mass matter right? for example the amu for copper is 63.5 and the amu for gold 197 and the specific heat capacity for each of the element is 376 and 126 respectively. Can you please explain why is that? why does it take less energy to heat up a greater mass? Thank you and sorry for bothering you Pieter master $\endgroup$
    – CountDOOKU
    Apr 18, 2019 at 13:49
  • $\begingroup$ i looked up heat capacity at wikipedia and it said 'specific heat capacity ... depends on degree of freedom' and then it adds on "larger the number of degrees of freedom available to the particles .... larger will be the specific heat capacity for the substance" if copper has more mass, isnt that means it harder to speed it up and shouldn't it then has a lower specific heat? $\endgroup$
    – CountDOOKU
    Apr 18, 2019 at 13:53
  • $\begingroup$ I had linked to Dulong and Petit's law. They discovered already two centuries ago that ratios of specific heats are as the ratios of their atomic masses. In the case of gold and copper 3:1. The reason is that each atom has the same thermal energy. $\endgroup$
    – user137289
    Apr 18, 2019 at 17:58
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This is probably a stupid question

This is not a stupid question. Look at the following graph from my statistical physics and thermodynamics course:

enter image description here

The graph shows molar heat capacity plotted against temperature along with different molecules. One notes here that different molecules have different (molar) heat capacities. When examining the graph further one maybe guesses or notes that more complex molecules seems to tend to have higher (molar) heat capacity. The explanation for this is that more complex molecule such as carbon dioxide $CO_2$ have more degrees of freedom to rotate and vibrate. Temperature can be thought of how much the particles or molecules move around while heat capacity can be thought of as how many different ways the molecules can vibrate, and then due to these vibrations "store" thermal energy. This explanation explains why the very simple nobles gases with almost none degrees of freedom to vibrate have low (molar) heat capacity and why carbon dioxide or water has high(er) (molar) heat capacity due to the more degrees of freedom to vibrate.

Edit: Note that the above discussion is primarily for gases. For solids the molar heat capacities (as the other answers have mentioned) is approximately the same for all solids, in accordance with Dulong–Petit law.

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The other answer correctly mentioned that the primary cause is different number of atoms, and the molar heat capacities are approximately the same, as per Dulong-Petit law. However attributing this to the heat capacity of specific atoms is not quite precise. In fact, mechanisms governing heat capacity in gases, metals, and isolators/semiconductors are quite different:

  • it is due to the kinetic energy of atoms in gases
  • it is primarily due to the conduction electron energy in metals
  • it is mainly due to phonons in non-conducting crystalline materials

Yet, one we treat heat capacity of insulators in terms of phonons (Debye model) and that of metals in terms of Landau quasiparticles, all the materials can be essentially viewed as gases (phonon gas, electron gas), and thus have gas-like heat capacity.

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    $\begingroup$ For metals the contribution of electron heat capacity is about two orders of magnitude smaller than that of the lattice, at room temperature. You may confuse it with thermal conductivity where indeed the free electrons may dominate. $\endgroup$
    – nasu
    Nov 12, 2021 at 12:12
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The other two answers are pretty much spot on. I will just summarize the concepts here.

First of all temperature is simply the degree of motion of the particles (nucleus and electron). For solid and liquid the motion is periodic vibration. For gas, the motion is random linear displacements. For any material, let's say solid, a certain temperature signifies that the particles of that material are vibrating at a certain frequency and amplitude.

Now, the specific energy is simply the energy needed to change that motion (increase or decrease). Since, the energy of motion is dependent on mass, environment etc. these factors will affect the specific energy as well. Since the atomic structure and density of different materials are different, their specific energy also differs.

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Because specifically the quantity of electrons of elements are different hence their ability to absorb heat energy differs before achieving a higher energy state.

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