The Third Law of Thermodynamics states that the entropy is zero when a system's temperature is absolute zero. A consequence of the Third Law is that we can determine the absolute entropy of a system.
Technically 0*ln(0) is undefined, but replacing the Boltzmann distribution in p*ln(p) one can see that the negative exponential will converge to 0 faster than the 1/T term it is multiplied by.
That's right. To keep things simple, I took a shortcut in claiming that p=0 was enough to make p ln p = 0. You're right that it is undefined, but approaches zero, so everything works out as claimed. In fact, you don't even need p to be a Boltzmann factor. It is generally the case that x ln x → 0 as x → 0⁺
What I'm wondering is this: To find an absolute entropy using the integral between 0 and the given temp, using the method discussed, ew are assuming the heat capacity to be a constant value. But we know heat capacity changed a load with temperature. Is there a neat way to correct the expression for this change? Thanks man
Your logic is sound, so clearly the heat capacity can **not** be constant! In fact, Cₚ behaves as Cₚ ~ T³ at low temperatures. We obtained a constant heat capacity for an ideal gas **only** if in the classical limit. (See this video, and the ones that follow it: ua-cam.com/video/5yEK0l3y08Y/v-deo.html ) Here's a video illustrating how the heat capacity (for solids) is roughly constant at high temperatures, but small at low temperatures: ua-cam.com/video/ZvwgqRmX4ks/v-deo.html
You could, if you did the heating at constant volume. (My example assumes constant pressure, although I didn't say so.) Typically, substances expand quite a bit as they are heated from 0 K, though, so heating at constant volume is not usually very practical.
Technically 0*ln(0) is undefined, but replacing the Boltzmann distribution in p*ln(p) one can see that the negative exponential will converge to 0 faster than the 1/T term it is multiplied by.
That's right. To keep things simple, I took a shortcut in claiming that p=0 was enough to make p ln p = 0. You're right that it is undefined, but approaches zero, so everything works out as claimed. In fact, you don't even need p to be a Boltzmann factor. It is generally the case that x ln x → 0 as x → 0⁺
What I'm wondering is this: To find an absolute entropy using the integral between 0 and the given temp, using the method discussed, ew are assuming the heat capacity to be a constant value. But we know heat capacity changed a load with temperature. Is there a neat way to correct the expression for this change? Thanks man
Great Video! But I have one question: If we take Cₚ to be constant (as for an ideal gas) the integral from 0 to T diverges. How can this be resolved?
Your logic is sound, so clearly the heat capacity can **not** be constant! In fact, Cₚ behaves as Cₚ ~ T³ at low temperatures.
We obtained a constant heat capacity for an ideal gas **only** if in the classical limit. (See this video, and the ones that follow it: ua-cam.com/video/5yEK0l3y08Y/v-deo.html )
Here's a video illustrating how the heat capacity (for solids) is roughly constant at high temperatures, but small at low temperatures: ua-cam.com/video/ZvwgqRmX4ks/v-deo.html
Can we use Cv for the same ?
You could, if you did the heating at constant volume. (My example assumes constant pressure, although I didn't say so.)
Typically, substances expand quite a bit as they are heated from 0 K, though, so heating at constant volume is not usually very practical.
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obrigado!