## Temperature Dependence of Entropy

Volume: $\Delta S = nR\ln \frac{V_{2}}{V_{1}}$
Temperature: $\Delta S = nC\ln \frac{T_{2}}{T_{1}}$

Curtis Tam 1J
Posts: 105
Joined: Thu Jul 13, 2017 3:00 am

### Temperature Dependence of Entropy

When we are calculating the entropy change for a reversible process, I understand that you use the equation deltaS=nRln(V2/V1) at constant temperature. For the equation for temperature dependence, deltaS=nCln(T2/T1), would that be used when the temperature is not constant since the final and intial temperature are different by a finite amount? I'm slightly confused by the wording of the course reader. It says: Calculate deltaS for change in temperature 1 and temperature 2 by summing infinitesimal increments (dS) at each temp dS=dqrev/T. So is the temperature dependent entropy change equation derived from an isothermal process? In other words, in what type of process do we use deltaS=nRln(V2/V1) and in what type of process do we use deltaS=nCln(T2/T1).

Sirajbir Sodhi 2K
Posts: 47
Joined: Sat Jul 22, 2017 3:00 am

### Re: Temperature Dependence of Entropy

The temperature dependent entropy change equation isn't exactly derived from the reversible isothermal work equation. But, it is similar in a sense: to get that equation, we assume that the change in temperature is infinitely small, that the process is isochoric (volume doesn't change). Then, we sum the infinitely small changes in entropy due to infinitely small changes in temperature due to heat transfer, and eventually, we are able to derive the equation you have written down. The other equation can only be used for the reversible, isothermal expansion of an ideal gas; in fact, it is derived from that work equation for that process.

Curtis Tam 1J
Posts: 105
Joined: Thu Jul 13, 2017 3:00 am

### Re: Temperature Dependence of Entropy

Sirajbir Sodhi 2K wrote:The temperature dependent entropy change equation isn't exactly derived from the reversible isothermal work equation. But, it is similar in a sense: to get that equation, we assume that the change in temperature is infinitely small, that the process is isochoric (volume doesn't change). Then, we sum the infinitely small changes in entropy due to infinitely small changes in temperature due to heat transfer, and eventually, we are able to derive the equation you have written down. The other equation can only be used for the reversible, isothermal expansion of an ideal gas; in fact, it is derived from that work equation for that process.

All right I kind of see what you're saying. I guess what I'm confused about is the idea of infinitesimal changes in temperature. Are you saying those small changes in temperature are what add up to a noticeable temperature difference between initial and final temperatures of a system over time?

Sirajbir Sodhi 2K
Posts: 47
Joined: Sat Jul 22, 2017 3:00 am

### Re: Temperature Dependence of Entropy

Yes. If you look at the P-V graph of a reversible isothermal expansion, P and V can change pretty heavily over a long period of time. That's true for temperature change in this case too.