# Entropy Equatio for a closed system

• Crazy Tosser
In summary, the conversation discusses the relationship between entropy and temperature in a closed system, with two different formulas being mentioned. The first formula is based on the assumption of reversible processes, while the second formula includes additional terms that take into account phase transitions. The conversation also touches on the constant heat capacity and its approximation in calculating entropy.

#### Crazy Tosser

I must have already been banned for spamming threads.

But oh well.

You know how change in entropy $$dS$$ of a closed system assuming reversibility of the processes = $$(\frac{dQ}{T})_{rev}=\frac{C_{p}dT}{T}$$

So when you try to find the actual entropy with respect to temperature, it's:

$$\displaystyle \int S_{T} = S_{0} + \int^{T}_{0} \C_{p}dT/T\$$

That's understandable. But that's my derivation. The text uses a slightly different formula:

$$S_{T} = S_{0} + \int^{T}_{0} \\frac{C_{p}dT}{T}\ + \sum \frac{\Delta H_{trans}}{T}$$

So my question 1 is: where did the $$\sum \frac{\Delta H_{trans}}{T}$$ come from?

And my question 2 is:
if we say that heat capacity $$C_{p}$$ is constant (it's not strictly constant, but with an approximation, we can call it a constant), we have:

$$\int \frac{C_{P}dT}{T} \sim C_{P} \int \frac{dT}{T}=$$
$$\sim C_{P} \ln{T}$$

Could the top eqn. be used as an approximation (part of an approximation, this is only a part of the equation)

As you can tell I am not that well-versed in Latex D=

Thanks

~Tosser

The ##\Delta H/T## terms come from the phase transitions from solid-to-liquid and from liquid-to-vapor (and any other phase transitions that occur, as, for example, from different crystal forms).