,
depends on the material, the amount of material, and the temperature. In
the limit where qV goes to zero
(so that ΔT also goes to zero) this ratio
becomes a derivative,
If we add heat to a sample of material, often the temperature will increase. (If we are at the temperature of a phase change, for example ice in water, the temperature will not change it will just melt some of the ice.) Away from a phase change adding heat will always give an increase in temperature. The amount of the temperature increase depends on how much heat was added, the size of the sample, the original temperature of the sample, and on how the heat was added. The two obvious choices on how to add the heat are to add it holding volume constant or to add it holding pressure constant. (There may be other choices, but they will not concern us.)
Let's assume for the moment that we are going to add heat to our sample
holding volume constant, that is, dV = 0. Let qV
be the heat added1 (the subscript, V,
indicates that the heat is being added at constant V). Also, let
ΔT be the temperature change. The
ratio, ,
depends on the material, the amount of material, and the temperature. In
the limit where qV goes to zero
(so that ΔT also goes to zero) this ratio
becomes a derivative,
We have given this derivative the symbol, CV, and we call it the "heat capacity at constant volume. Usually one quotes the "molar heat capacity,". (1)
We can rearrange Equation 1 as follows,. (2)
Then we can integrate this equation to find the heat involved in a finite change at constant volume,. (3)
If CV is approximately constant over the temperature range then CV comes out of the integral and the heat at constant volume becomes,(4)
Let us now go through the same sequence of steps except holding pressure constant instead of volume. Our initial definition of the heat capacity at constant pressure, Cp becomes,. (5)
The analogous molar heat capacity is,. (6)
Equation (6) rearranges to,. (7)
which integrates to give,, (8)
When Cp is approximately constant the integral in Equation (9) becomes. (9)
Very frequently the temperature range is large enough that Cp cannot be regarded as constant. In these cases the heat capacity is fit to a polynomial (or similar function) in T. For example, some tables give the heat capacity as,. (10)
where α , β , and γ are constants given in the table. With this temperature-dependent heat capacity the heat at constant pressure would integrate as follows,, (11)
. (12a, b)
Occasionally one finds a different form for the temperature dependent
heat capacity in the literature,
. (13)
When you do calculations with temperature dependent heat capacities
you must check to see which form is being used for Cp.
1. We are using the convention that q will always designate heat absorbed by the system. q can be positive or negative and the sign indicates which way heat is flowing. If q is positive then heat was indeed absorbed by the system. On the other hand, if q is negative it means that the system gave up heat to the surroundings.
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