A heat engine takes 0.350 mol of a diatomic ideal gas around the cycle shown in the pV-diagram o

A heat engine takes 0.350 mol of a diatomic ideal gas around...

Key Concepts

Thermal Efficiency and Carnot Cycle

Thermal efficiency measures the effectiveness of a heat engine in converting absorbed heat into work, and it is defined as the ratio of the work output to the heat input. The Carnot cycle, which represents an idealized reversible cycle operating between two heat reservoirs, provides the maximum theoretical efficiency given by 1 - (T_min/T_max). Comparing an actual engine's efficiency to that of a Carnot engine offers insights into the real-world limitations imposed by irreversibilities and losses.

First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation and declares that the change in a system's internal energy is equal to the heat added to the system minus the work done by the system. This principle forms the backbone of thermodynamic analysis, enabling the characterization of energy transfers during various processes within a cycle.

Isobaric Process

An isobaric process occurs at a constant pressure. This type of process is significant because it simplifies the work calculation, as work done is directly proportional to the change in volume. In many practical situations, such as in certain stages of heat engines, the assumption of constant pressure allows for straightforward application of thermodynamic principles.

Adiabatic Process

An adiabatic process is characterized by the absence of heat exchange with the surroundings. In this process, any change in the system's internal energy is solely due to the work done on or by the system. The relationship between pressure and volume during an adiabatic process is given by pV^? = constant, where ? (gamma) is the specific heat ratio.

Isochoric Process

An isochoric process is one in which the volume remains constant. In such a process, no work is done because work is the product of pressure and change in volume. Any heat added to or removed from the system directly changes the internal energy, making this process a direct application of the first law of thermodynamics without the work term.

Thermodynamic Processes

Thermodynamic processes describe how a system changes state, and each type of process (constant volume, constant pressure, adiabatic, etc.) has specific characteristics. Understanding the nature of these processes is fundamental when analyzing cycles, since each process dictates how variables like pressure, volume, and temperature evolve, and they determine how energy in the form of heat and work is exchanged.

pV Diagram

A pV diagram is a graphical representation that plots the pressure of a system against its volume. It provides a visual depiction of different thermodynamic processes and cycles, such as those occurring in heat engines, allowing one to analyze processes like expansion and compression, and calculate work done as the area under or enclosed by the process curves.

Ideal Gas (Diatomic)

An ideal gas is a theoretical construct in which the gas molecules do not interact except through elastic collisions, and it obeys the ideal gas law. For diatomic gases, additional degrees of freedom due to rotation (and sometimes vibration) affect properties such as heat capacity and the specific heat ratio, ?. This ratio plays a crucial role in determining the behavior of diatomic gases during thermodynamic processes like adiabatic expansion or compression.

Heat Engine

A heat engine is a device that converts thermal energy into work by undergoing a cyclic process where heat is absorbed from a high-temperature source and partially converted into work, while the remaining energy is exhausted to a lower-temperature reservoir. This fundamental concept is central to thermodynamics and engineering, illustrating the practical limits on energy conversion imposed by the second law of thermodynamics.

Transcript

00:02 Okay, so this problem asks a lot of questions, and so it's basically talking about an ideal gas going around a particular cycle, and the information is communicated through a pv diagram.

00:20 So i went ahead and wrote down pv diagram, the temperatures that are given, and then i'm sure i'm going to forget, realize i forgot to write some things down.

00:31 I'm doing that right now.

00:34 We're given that the pressure at 1 is 1 atm, and we're asked to find so many quantities.

00:43 So i went ahead in rotown, like, all the numbers that we're going to use, so gamma is 1 .4, then we can get the heat capacity, constant volume, using this formula from the book, r over gamma minus 1.

00:54 And our standard constant, the conversion that 1 atmosphere is 1 .03 to the 5 pascals.

01:02 Si units and that there's 0 .35 moles.

01:07 So i'm not gonna write out all the calculations that i do, but you can just refer to this page for what to type into a calculator if there's some question.

01:19 Okay, so now to start the problem, so they ask you to find p and b everywhere.

01:27 So let's start with one.

01:31 So this, oh, let me erase that.

01:36 Okay.

01:38 And why not just use a different color to label this a? this whole sheet will be dedicated today.

01:48 So you want to find p and v at 1.

01:51 So start with ez first.

01:54 P at 1 is equal to 1 .03, 10 of the 5 pascal.

01:59 I was just reading off the graph.

02:00 And then v is found by doing nrt1 divided by p -o -1.

02:09 All those numbers should be up for you on the previous sheet.

02:14 So if you plug that into a calculator, maybe you'll get what i got, get what i got, which is that.

02:22 Great.

02:24 And for 2, v is the same.

02:28 So that's 2 is when the, yeah, it just goes up in pressure.

02:32 So to find v, you need to do acknowledge that.

02:37 If v stays the same, then the quantity p over t stays the same.

02:46 So considering going from 1 to 2, oops, then p1 over t1 equals p2 over t2, and then if you're confused, it'll probably, might drive you crazy that this is not prettily labeled.

03:04 So i will go ahead and label that nicely.

03:08 And so just to kind of motivate for like why i did that, if pv, so we have the ideal gas law, right, pv is nrt.

03:17 So i like to like circle what are the same.

03:20 And so in that case, n's going to say the same are the constant.

03:23 So it stays the same.

03:24 And then if you put them on one side of the equation like this, then you get v over nr is p over t.

03:31 So if this, stays constant then p over t has to stay constant that's why you know that it's the same and so if you solve this for p2 you got it's equal to t2 over t1 multiplied by p1 and maybe you'll get what i got the ratio between the temperatures is two so you just want to multiply the p at 1 it times 2 so that's 2 .026 times 2 10 to the 5.

04:05 Great.

04:06 And you're going to do, so now to consider three, you want to consider the path from one to three because that's the simplest.

04:12 Way to do that because you don't have to worry about, well, i guess you'll see why it's simple.

04:19 Relatively simple.

04:22 So you can kind of do the same thing and figure out what stays the same.

04:25 So from 3 to 1, p is the same.

04:30 Therefore, v1 over t1 equals, v3 over t3 and you can just make the same argument that we made over here and then from that you got that v3 is t3 over t1 times v1 and that's 0 .014 meter cubed um so p and then just to be clear p3 equals p1 um and there's so many answers i'm just not going to box them.

05:07 On to sheet b.

05:10 I think this one will get its own sheet too.

05:13 Oh no, the top's cut off.

05:15 Okay, there's sheet b.

05:19 So now we're considering like the different energy measurements for each process.

05:30 So for all these, we want to use this kind of relationship between the, oh, that was a nice triangle, delta e, internal.

05:38 Equals the minus work done by the gas plus the heat added to the gas.

05:48 This one is always going to be from n, c, v, delta, t.

05:55 This one's like always p delta v.

06:04 And then q, you always have to just deduce from this equation.

06:12 So it doesn't have like its own independent way to measure it.

06:23 So now we want to consider the process of one to two.