Entropy in the Real World: Engines Figure 20-27 shows a...

Entropy in the Real World: Engines Figure 20-27 shows a reversible cycle through which 1.00 $\mathrm{mol}$ of a monatomic ideal gas is taken. Assume that $p=2 p_{0}, V=2 V_{0}, p_{0}=$ $1.01 \times 10^{5} \mathrm{Pa},$ and $V_{0}=0.0225 \mathrm{m}^{3}$. Calculate (a) the work done during the cycle, (b) the energy added as heat during stroke $a b c,$ and $(c)$ the efficiency of the cycle. (d) What is the efficiency of a Carnot engine operating between the highest and lowest temperatures that occur in the cycle? (e) Is this greater than or less than the efficiency calculated in $(\mathrm{c}) ?$

Entropy in the Real World: Engines
Figure 20-27 shows a reversible cycle through which 1.00 $\mathrm{mol}$ of a monatomic ideal gas is taken. Assume that $p=2 p_{0}, V=2 V_{0}, p_{0}=$ $1.01 \times 10^{5} \mathrm{Pa},$ and $V_{0}=0.0225 \mathrm{m}^{3}$. Calculate (a) the work done during the cycle, (b) the energy added as heat during stroke $a b c,$ and $(c)$ the efficiency of the cycle. (d) What is the efficiency of a Carnot engine operating between the highest and lowest temperatures that occur in the cycle? (e) Is this greater than or less than the efficiency calculated in $(\mathrm{c}) ?$

Key Concepts

Thermodynamic Cycle

A thermodynamic cycle is a series of processes that a system undergoes returning to its original state, resulting in net work output or input over one complete cycle. It is typically represented on a pressure-volume (p-V) diagram, where the area enclosed by the cycle indicates the net work done. This concept is fundamental in analyzing and designing engines and refrigerators.

Reversible Process

A reversible process is an idealized, quasi-static process that proceeds infinitely slowly so that the system is in thermodynamic equilibrium at every stage. In such a process, the entropy change of the universe is zero, and it represents the maximum theoretical efficiency for heat engines. It serves as a standard for comparing real processes to ideal limits.

Work in Thermodynamic Cycles

Work in a thermodynamic cycle is the energy transferred by the system to its surroundings (or vice versa) as it undergoes expansion and compression. This work is represented by the integral of pressure with respect to volume (?p dV) over the cycle, and it is graphically depicted as the area enclosed by the cycle on a p-V diagram.

Heat Transfer in Thermodynamic Processes

Heat transfer in thermodynamic processes involves the flow of energy due to temperature differences between a system and its surroundings. During a cycle, energy is added as heat during certain processes and rejected in others, and this exchange of energy is governed by the first law of thermodynamics, linking heat, work, and internal energy changes.

Heat Engine Efficiency

Heat engine efficiency is the measure of how well an engine converts the heat energy it receives into work. It is defined as the ratio of the net work output to the heat input. This concept is central in thermodynamics, as it determines the performance and viability of energy conversion systems.

Carnot Cycle Efficiency

The Carnot cycle efficiency represents the maximum possible efficiency that any engine operating between two given temperatures can achieve. It is derived from a reversible Carnot cycle and is given by 1 minus the ratio of the cold reservoir temperature to the hot reservoir temperature. This ideal efficiency serves as an upper bound for all real heat engines.

Ideal Gas Laws

The ideal gas laws describe the relationship between pressure, volume, and temperature for an ideal gas. These laws, encapsulated in the equation PV = nRT, where n is the number of moles, are crucial for analyzing processes involving gases, particularly in determining changes in state within a thermodynamic cycle.

Transcript

00:01 So in this problem, we are given a process of gas goes through.

00:05 We know the values p, v and p .0 and v0 and n, and we are supposed to find a few things.

00:12 The first thing is the amount of work.

00:14 If you remember, the work is equal to the area of this graph.

00:19 Now, p .0 and v0 are given, and we know that the cycle goes another p .0 and another v0.

00:30 So this is because we know that p is equal to 2p0, which means this whole length is 2p0, and this whole length is 2v0.

00:41 So we know that this side of the of the rectangle is p .0 and this side of the rectangle is v0.

00:48 So the area is simply their product.

00:50 So that's p .0 v0.

00:53 And this, when you multiply these two numbers simply turns out to be 2 .27 into 10 par 3 joules.

01:00 So that's 2 .27 kilojoules.

01:03 Now that you know this, we want to find out the amount of heat added.

01:10 So now, in the process ab, in this process, remember that this is at constant volume.

01:19 So the heat is ncv delta t.

01:24 And in this process, which is at constant pressure, heat is ncp delta t.

01:31 And that is it.

01:33 We'll use our formula.

01:37 N is one, so we can simply ignore it for now.

01:40 We know cv and cp for a monoatomic gas.

01:43 Cv is 3x2r, so that's 3 by 2r.

01:47 T is and delta t is simply t b minus ta.

01:54 For the second case, it's cp is 5 by 2r and the delta t is tc minus tb.

02:04 And that is it we'll have to use this in our value now remember that we know we don't know t and t v but what we can do is simply pull out t a outside now we know we can see that 3 by 2 r and if we pull t a out of out of this and write t a and t b by t a minus 1 now if you remember pv is equal to n rt which means t2 by t1 is p2 v2 by p1 v2 by p1.

02:42 So here, remember that this is basically n rt because n is 1, so we can simply multiply it by n.

02:51 So this is p .0 v0, that is p .a, va.

02:54 And this is simply pb vb by pava, which is p00 v0 minus 1.

03:04 Now the volumes at b and they are same, but the volume at b, the pressure at b, is twice...

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