A monatomic ideal gas is taken around the cycle shown in Fig. 120.42 in the direction shown in t

A monatomic ideal gas is taken around the cycle shown in...

Key Concepts

Ideal Gas Law

The Ideal Gas Law, expressed as pV = nRT, is a fundamental thermodynamic equation that relates the pressure, volume, and temperature of an ideal gas through the amount of substance present. This law is crucial for understanding how gases behave under various conditions and is used to analyze the state changes during thermodynamic processes.

First Law of Thermodynamics

The First Law of Thermodynamics is the principle of energy conservation for thermodynamic systems. It states that the change in internal energy of a system (?U) is equal to the heat added to the system (Q) minus the work done by the system (W). This relationship, ?U = Q - W, is central to analyzing energy transfers during individual processes and cycles.

Internal Energy of a Monatomic Ideal Gas

For a monatomic ideal gas, the internal energy depends solely on the temperature and is given by U = (3/2)nRT. This concept is important because it implies that any change in the internal energy of such a gas is directly related to a change in temperature, regardless of the path taken during the process.

Work in Thermodynamic Processes

Work in thermodynamic contexts is the energy transferred when a force moves the boundary of the system, often calculated as the integral of pressure with respect to volume (W = ?p dV). The work done is path-dependent, meaning the amount of work performed during a process depends on how the state variables change along the path in the p-V diagram.

Heat Transfer in Thermodynamics

Heat transfer (Q) involves the exchange of energy between a system and its surroundings due to temperature differences. In thermodynamic processes, determining the amount of heat added or removed from the system is essential for applying the first law and understanding changes in internal energy.

Efficiency of a Thermodynamic Cycle

The efficiency of a thermodynamic cycle is the ratio of the net work output of the cycle to the heat input. It is a measure of how effectively a system, such as a heat engine, converts heat energy into work. Efficiency is defined as ? = W_net/Q_in, and its analysis is critical in evaluating the performance of cyclic processes.

Transcript

00:01 Whenever analyzing a cycle with an ideal gas, there are some tools that you want to start to write down.

00:09 The first is the ideal gas law, simply written as p pressure times volume, its number of moles, times gas constant, times temperature.

00:25 Temperature being in kelvin.

00:28 The second tool is the first law of thermodynamics, which says, that basically energy is conserved.

00:39 So a change in internal energy is one form of energy, and it is equal to the heat absorbed when positive or negative when expelled, minus the work done by the gas.

00:58 So one thing that is helpful is also that the change in internal energy of a gas if it's ideal, can be calculated from the product of three halves, well, sorry, cv times the number of moles, times the change in temperature gives the change in internal energy.

01:27 And we get cv based on whether it is a monotomic gas, a diatomic ideal gas, etc.

01:34 For a monotomic gas, we know that that cv is three halves times r, r being the gas constant.

01:46 So we know immediately that cv is, what cv is.

01:53 We'll leave it at that.

01:56 The other thing to recognize is what type of device that you have, which you can get from looking at the pv diagram.

02:06 If the arrows are such that they have an increase in volume, that's bigger than the decrease in volume, then we know it is basically a heat engine.

02:21 If the arrows run in reverse, it is a refrigerator, heat pump, or a chiller of some sorts.

02:29 So we definitely have a heat engine in this particular example, and we can write that its efficiency is the ratio of the work output divided by the heat absorbed, and then you often multiply that by 100%, just to have kind of a basic way to talk about that efficiency.

02:57 So what we are going to be doing is we're going to be calculating the parts of the first law for each of the processes, a to b, b to c, and c to a.

03:07 And then we can step back and see what the three p parts of the first law of thermodynamics are for the entire cycle.

03:18 And there should be some things that are fairly easy to check.

03:23 So we'll kind of make a table and leave some room for some calculations.

03:29 But a to b, there are two things that are easy to find.

03:34 One is the work done is going to be involved.

03:38 With the pressure times the change in volume, or the area under the graph is a way of thinking about it.

03:48 The delta u is easy to calculate.

03:54 It is three halves r times n delta t.

04:07 And i'll write that slightly differently, which you'll see why a little bit three halves n -r -t and then q is the weird one.

04:21 It has to come well it's not really weird but there's no way to always calculate it directly.

04:28 It must come from the delta u plus the work done by the gas, rearranged from the first law.

04:36 So we'll kind of set up some regions to do these calculations and we'll look at a to b first.

04:43 The pressure remains three times 10 to the 5th pascal's.

04:50 It is a constant pressure process, and 0 .3 cubic meters will give us the work done in jewels.

05:07 This is positive because the gas is expanding and can actually push against something to do mechanical work.

05:18 The delta u, now notice that we do not have.

05:21 The temperatures at our points, nor can we find those because we don't know the number of moles.

05:27 So we're going to leave the nr delta t grouped together and use the ideal gas law to work out the full product in our delta t.

05:43 So delta u is going to be three halves, and we're going to have a difference than in pressure times volume, which is the other side of the ideal gas.

05:57 Law.

06:00 So between a and b, we have three halves times pressure b, volume b, minus pressure a, volume a.

06:13 And working that out, that winds up to be 1 .35 times 10 to the 5th joules.

06:26 And finally, we have to add together our work and our change in internal energy to get q and we get 2 .25 times 10 to the 5th joules.

06:44 So we simply want to add our two columns to the left of q to get what q is.

06:52 So that is a process where the temperature increased and heat was absorbed.

06:59 So it's sometimes good to show that q absorbed.

07:08 All right, let's go b to c.

07:14 Work is zero.

07:15 Zero.

07:17 That is easy because no change to the volume.

07:26 So the gas actually has to expand to do positive work or contract while doing negative work...

Please give Ace some feedback