(a) What is the efficiency for a reversible engine operating...

(a) What is the efficiency for a reversible engine operating around the indicated cycle, where $T$ is temperature in K and $S$ is the entropy in joules/K? FIGURE CAN'T COPY. (b) A mass $M$ of a liquid at a temperature $T_1$ is mixed with an equal mass of the same liquid at a temperature $T_2$. The system is thermally insulated. If $c_p$ is the specific heat of the liquid, find the total entropy change. Show that the result is always positive.

(a) What is the efficiency for a reversible engine operating around the indicated cycle, where $T$ is temperature in K and $S$ is the entropy in joules/K?
FIGURE CAN'T COPY.
(b) A mass $M$ of a liquid at a temperature $T_1$ is mixed with an equal mass of the same liquid at a temperature $T_2$. The system is thermally insulated. If $c_p$ is the specific heat of the liquid, find the total entropy change. Show that the result is always positive.

Key Concepts

Reversible Thermodynamic Cycles

A reversible thermodynamic cycle is an idealized series of processes in which the system and its surroundings can be returned to their original states without any net change in entropy. This idealization is used to determine the maximum possible efficiency of a heat engine, as all reversible processes are lossless and have no internal irreversibilities.

Carnot Efficiency

Carnot efficiency defines the maximum efficiency that any heat engine operating between two heat reservoirs can achieve, and it is given by the formula 1 - (T_low/T_high). This result stems from the reversible (Carnot) cycle, where the efficiency is determined solely by the temperatures at which heat is absorbed and rejected.

Entropy and the Second Law

Entropy is a fundamental concept in thermodynamics that quantifies the disorder or randomness in a system. The Second Law of Thermodynamics states that, for any spontaneous process in an isolated system, the total entropy must increase. This principle underlies the analysis of both reversible cycles (zero net entropy change) and irreversible processes (positive net entropy change).

Entropy Change Due to Mixing

When two bodies at different temperatures are combined in an insulated system, the process is irreversible and results in a net increase in entropy. The calculation of this entropy change typically involves the specific heat of the material and the logarithmic functions of the initial and final temperatures, which capture how the dispersed energy leads to greater randomness in the system.

Transcript

00:01 In this problem, we have a real irreversible engine, not a reversible one.

00:05 Remember, ask some questions about what is the entropy and work and other things related and how they relate to a reversible engine? so let's start with the first question.

00:19 That's asked for the change of entropy of the universe.

00:23 Notice whatever's going on in here in this part of the system that's going through a cycle.

00:30 And think of the carnal cycle.

00:33 And delta s for cycle, reversible, reversible does not matter.

00:38 Delta s for a cycle is zero.

00:43 Entropy is a state function.

00:45 All it matters is where you are.

00:46 If you end up where you began, then delta s is zero.

00:50 So all we got to worry about is what's going on with these reservoirs and the heat transfers into and out of the reservoirs.

00:58 So reservoirs, meaning they don't change in temperature, so i think of these isothermal transfers.

01:04 So the first one, i'm going to be taking qh out, minus qh over th plus.

01:12 Now, now, qc is going in, qc over tc.

01:16 So that is the answer to part a.

01:20 Part b, they want to know, is this greater than equal to zero? now, just from the second law tells you that it's going to be greater than zero.

01:29 But let's see if we can understand a little more about that.

01:34 What's happening is the following.

01:36 Think about people walking around a room along the perimeter.

01:41 And you tell them, you add heat by telling them to speed up.

01:47 Now, they're still along the perimeter.

01:49 So it's very easy to tell them to slow down, they're still around the perimeter.

01:52 You've got the same order, or if you like the same disordered state from one time to another, and you can easily get back.

01:58 So the entropy change is zero between those states.

02:02 Because if you like, the disorder is not changed.

02:06 Now, though, irreversible process.

02:10 This qc, it's qc in this particular case, it's got an added aspect to it.

02:17 It's got a disorder piece to it.

02:19 In addition, that's the guy who would tell people, you know, you have the one part that says speed up, he's busy saying, now move randomly into the room.

02:29 You know, i'm giving you energy to move into the room.

02:33 You tell them to slow down, you're not going to get back to that nice motion along the perimeter, that state of order or that state of disorder, you've got a much more disordered state.

02:44 So that's why the entropy of the universe increases, because you've got this added aspect, in this case for qc.

02:53 Okay.

02:54 Part c, they're looking for the relationship between the irreversible w and the reversible.

03:01 So delta u is equal to zero.

03:05 We're just looking at this part now, just the working aspect, you know, the gas, whatever the system is here, not its interaction with the rest of the universe.

03:16 This is equal to zero.

03:18 It's a cycle, right? as i talked about with delta s just a minute ago.

03:23 So this is qh minus qc.

03:28 This is the first law.

03:31 So i get that w is equal to qh as qc.

03:37 This is true reversible, irreversible? it does not matter.

03:40 Now, i just mentioned that qc has got this added aspect...

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