The Physics Philes, lesson 121: Mountains Out of Mole Hills
Is it hot in here? Yes! Because we’re continuing our discussion of heat!
Last week we learned about specific heat. We generally measure specific heat as a mass per change in temperature. However, sometimes we might want to use the number of moles of a substance, rather than the substance’s mass.
Whoa whoa whoa…mole? Like an animal?
No. Don’t be silly. A mole is the number of molecules in a pure substance. Not just any number. A mole of any pure substance has the same number of molecules. Specifically, that number is 6.022 x 10^23 molecules per mole. A mole of gold has 6.022 x 10^23 molecules of gold. A mole of aluminum has 6.022 x 10^23 molecules of aluminum. A mole of water has 6.022 x 10^23 molecules of water. You get the point. It’s a handy unit of measurement.
We can express specific heat in terms of molar mass M. If you’ll remember, our expression for the heat required to affect some change in temperature for a mass m is Q = mcΔT, where c represents the specific heat of a particular material. We can’t just replace the mass of the substance m with the molar mass M, however. The total mass of a material is equal to the molar mass M times the number of mole n: m = nM. Now that we know that relationship, we can substitute it into the initial equation, and we get this:
Q = nMcΔT
Inside this equation we have something new. The product Mc shall hereby be dubbed the molar heat capacity or molar specific heat C. Now we have an equation that tells us the heat required to change the temperature of some number of moles of a substance:
Q = nCΔT
Like specific heat c, this is determined experimentally and is different for every substance. Not only do these values need to be determined experimentally, but you have to determine them differently for solids and gases. To measure C for a solid, it’s easier to do the experiment under constant pressure. For gases, it’s easier to measure C under constant volume. For any given substance, these two values of C will be different, but they can be related like this:
where Cp is the molar specific heat at constant pressure, Cv is the molar specific heat at constant volume, and R is the gas constant that has the same value for all gases, 8.31 J/mol⋅K.
These values are different for the same substance because of what it means for the work done by the system. If the system can expend while heat is added – like we would see if the pressure is constant – there is additional energy exchanged through the work done by the system on its surroundings. However, when we hold the volume constant, no work is done.
There is one weird thing that becomes obvious as we measure specific heats. The molar specific heat for most elemental solids are around the same value, about 25 J/mole⋅K. This is called the rule of Dulong and Petit, and it’s the foundation of a pretty important idea. On a per atom basis, the amount of heat it takes to raise the temperature of these elements a certain amount is the same even though the actual masses of the atoms are different, sometimes very different. This is so because, as you’ll remember, a mole of any substance has the same number of molecules. So the amount of heat required to cause an increase in temperature only depends on how many atoms are in the sample.
See? I told you that moles is a pretty handy unit.
And that is it for specific heats! Next time we’ll discuss what it takes to get some substance to transform from one state of matter to another.