Entropy and Heat Conduction

Question

For an Intro. Thermal Physics course i am taking this year, I had a simple problem which threw me off-guard, I would appreciate some input to see where i am lacking. The problem is as follows:

Does the entropy of the substance decrease on cooling? If so, does the total entropy decrease in such a process? Explain.

Here is how i started this:

->Firstly, for a body of mass m and specific heat, c(assuming it is constant) the heat absorbed by the body for an infinitesimal temperature change is $dQ=mcdT$.

->Now if we raise the temperature of the body from $T_1$ to $T_2$, the entropy change associated with this change in the system is $\int_{T_1}^{T_2}mc\frac{dT}{T}=mcln\frac{T_1}{T_2}$. This means the entropy of my system has increased. Up to this was fine.

I face difficulty in the folowing:

<*>Is this process, the act of heating this solid, a reversible or an irreversible one? Now, I know that entropy is a state variable, so even if it was irreversible, so to calculate the entropy change for the system during this process we must find a reversible process connecting the same initial and final states and calculate the system entropy change. We can do so if we imagine that we have at our disposal a heat reservoir of large heat capacity whose temperature T is at our control.

We first adjust the reservoir temperature to $T_1$ and put the object in contact with the reservoir. We then slowly (reversibly) raise the reservoir temperature from $T_1 to T_2$. The body gains entropy in this process, the amount i have calculated above.

According to the main problem, if i were to reverse this process and slowly lower the temperature of the body from $T_2$ to $T_1$ wouldn't the opposite were to happen? i.e. the body loses entropy to the reservoir, the same amount as calculated above, but different signs?

<*> From above discussion, can i say that the net entropy of the system+surroundings is zero? Had it been a reversible process then from the second law i know it would've been zero, even if it is irreversible, as long as i connect the same two states with a reversible path, the net still comes out to be zero.

Am i right to think of it as such? I had this problem of discerning which is reversible/irreversible for a while.

Answer 1

Your reasoning appears to be sound.

The only issue I see is when you talk about raising the temperature of the reservoir. If it is a reservoir its temperature can't technically change. Instead, you would theoretically need to place the body in contact with an infinite series of reservoirs, each differing from the prior by $dT$. The same goes for the reverse process to bring the body back to temperature $T_1$.

Then each process is reversible, the first decreasing the entropy of the body and the second increasing the entropy of the body by the same amount. The two processes together then constitute a reversible cycle.

Hope this helps.

Answer 2

The original equation you wrote assumes that, rather than putting body in contact with a reservoir at T2 for the entire process, it is put in contact with a sequence of reservoirs at gradually varying temperatures running from T1 to T2. This would be equivalent to the "reversible" process you described of gradually changing the temperature of the reservoir from T1 to T2 (although, to bring this about, you would have to implicitly be using a reservoir of finite heat capacity, rather than an ideal constant temperature reservoir of infinite heat capacity). But the question then arises, "how was the temperature of the reservoir gradually changed?" To do this reversibly, you would still need to use a supplementary sequence of reservoirs at gradually varying temperatures to transfer heat to the finite heat capacity reservoir. So the net result would be the same for the system, and also for the combined surroundings consisting of an intervening finite heat capacity reservoir and a sequence of ideal infinite heat capacity reservoirs of gradually varying temperatures.

If the actual irreversible process were done by putting the reservoir into contact with an ideal constant temperature reservoir of T2 throughout the process, the change in entropy of the body would still be the same. But the change in entropy of the ideal reservoir would be higher, at $$-\frac{mc(T_2-T_1)}{T_2}$$

I leave it to you to work out the equation for the reservoir entropy change in the case of cooling.