Molar heat capcity for any solid is 3R which is approximately 24J/kg-K.
But specific heat of aluminium is 900J/kg-K.
If the molar heat capacity is M times more than specific heat capacity (M=molar weight) the values become incorrect.
Where am I going wrong?
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2 Answers
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You are going wrong in the Units. The Molar Heat Capacity of any solid is approximately 24J/K-mol. Molar mass of Aluminium is around 27 grams. i.e 0.027 Kg.
Specific heat capacity of Aluminium hence is (Approximately), 24J/K-(0.027)Kg.
Which comes out to be 889J/K-Kg. Which is good approximation to 900J/Kg-K, Given the generality of dulong-petit law, and cumilative effect of both our approximations.
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$\begingroup$ I suppose that numerically, you'd need a substance with a molar mass of over 1 kg for the molar heat capacity in $\text{J/K}\cdot\text{mol}$ to come out less than the specific heat capacity in $\text{J/K}\cdot\text{kg}$. But that'd have to be a pretty hefty molecule. $\endgroup$ Dec 10, 2020 at 1:56
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It depends on how you measure it. Either type of heat capacity is just how much energy it takes to raise some given amount of a substance by a given unit of temperature. The exact amount is directly proportional to amount of substance that your units consider. Specific heat capacity is measured per unit mass, while molar heat capacity is measured per mole. Specific heat capacity is just a broad category of how to define an "amount" of something, but not particular units. Molar heat capacity is specifically in the unit of moles, although it belongs to a different broad category of how to define an "amount" of something that also includes other units proportional to moles.*
A mole of any chemical substance is always going to be more than 1 gram, since the smallest atom, a hydrogen-1 atom, is 1.007825032241(94) atomic mass units (a.k.a. daltons). Thus, the specific heat capacity in J K-1 g-1 will always be higher than the molar heat capacity in J K-1 mol-1, unless your dealing with some exotic substance like an electron gas or maybe neutronium if binding energy drops the mass of the neutrons or something else where a mole of substance is less than 1 gram. On the other hand, the specific heat capacity in J K-1 kg-1 is usually higher than the molar heat capacity in J K-1 mol-1, with the exception being when you're dealing with huge molecules of mass greater than 1000 amu(a.k.a. da) or some other situation where the molar mass is greater than 1 kg.
*For a metal like aluminium, you could have an atomic heat capacity, which would be how much energy is required to increase the average translational kinetic energy of the atoms, and which would be proportional to the molar heat capacity because a mole is just Avogadro's number of atoms (or molecules, etc. in other cases).
Since since translational kinetic energy, which is directly proportional to temperature, is where most of the heat energy put into a monatomic substance like aluminium goes, the heat capacity is similar for all such substances. (You should be aware that there are actually 2 different relevant heat capacities, one for constant pressure and one for constant volume(i.e. constant density), but I think that constant pressure is the most useful and most commonly used one for solids and liquids by far.) If all of the energy you put in were going into just moving the atoms, as in an ideal monatomic gas, then the heat capacity at constant pressure (CP) would be 20.79 J K-1 mol-1; however, real monatomic solids and liquids also spend some of the heat energy you put in on other things like expanding and maybe some kind of lattice vibrations that don't increase temperature (I don't know) or exciting electrons, just like how molecular substances spend some heat energy on intramolecular rotations and vibrations and plasmas spend some heat energy on exciting electons and ionizing atoms. Thus, the real CP will always be higher than 20.79 J K-1 mol-1 (or approximately equal to it within the precision of that number). For illustration, some heat capacities for aluminium going up to high temperatures from Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (105 Pascals) Pressure and at Higher Temperatures by Richard A. Robie and Bruce S. Hemingway for U.S. Geological Survey Bulletin 2131, published in 1995, are given below. The only phase change is aluminium melting at 933.5 K — no crystal structure changes — and, as the name implies, the pressure is 1 bar (almost 1 atm).
T/K--------CP (J K-1 mol-1)
298.15-----24.21
300--------24.25
400--------25.76
500--------26.84
600--------27.90
700--------29.12
800--------30.57
900--------32.30
933.5------32.96 (solid)
933.5------31.75 (liquid)
1000-------31.75
1100-------31.75
1200-------31.75
1300-------31.75
1400-------31.75
1500-------31.75
1600-------31.75
1700-------31.75
1800-------31.75
These temperature-dependent changes in heat-capacity have to do with how temperature affects the various places that the heat energy you put in can go other than translational motion, but notice that all these numbers are decidedly above the theoretical minimum for ideal gasses, and also, as it happens, the theoretical estimation you gave. Honestly, I don't know where you got your 24 J K-1 mol-1 estimation because I'm no expert in the heat capacities of solids, but I don't doubt too much that it's a good estimation, probably one that takes into account some particular place other than translational motion that kinetic energy can usually go in solids.
