20.41. You build a heat engine that takes 1.00 mol of an...

20.41. You build a heat engine that takes 1.00 mol of an ideal diatomic gas through the cycle shown in Fig. 20.25 . (a) Show that segment ab is an isothermal which segment(s) of the cycle is heat absorbed by the gas? During which segment(s) is heat rejected? How do you know? (c) Calculate the temperature at points $a, b,$ and $c .$ (d) Calculate the net heat exchanged with the surroundings and the net work done by the engine in one cycle. (e) Calculate the thermal efficiency of the engine.

Key Concepts

Heat Engine

A heat engine is a device that converts thermal energy into mechanical work by undergoing a cyclic process. In such engines, the working substance (often modeled as an ideal gas) absorbs heat from a high-temperature reservoir, performs work, and rejects some heat to a low-temperature reservoir in order to complete the cycle.

Thermodynamic Cycle

A thermodynamic cycle consists of a series of processes that return a system to its initial state. In each cycle, the net change in internal energy is zero, and the cycle’s performance is evaluated by the work done and heat exchanged during the cycle. Graphical representations like PV diagrams are often used to illustrate these cycles.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. It is central to analyzing the behavior of gases during various thermodynamic processes, including those in heat engine cycles, making it a crucial tool for predicting state variables at different points in the cycle.

Isothermal Process

An isothermal process is one that occurs at a constant temperature. For an ideal gas undergoing an isothermal change, the product of pressure and volume remains constant, and the work done can be quantified by integrating the pressure with respect to volume. Recognizing a process as isothermal allows for specific simplifications and analytical approaches to calculate work and heat transfer.

Heat Transfer in Thermodynamic Processes

Heat transfer in a thermodynamic process refers to the energy that enters or leaves the system as heat. In a cyclic process, different segments may involve either absorption (where the system gains heat) or rejection (where the system loses heat). Identifying which segments involve positive or negative heat transfer is crucial for analyzing the engine’s performance.

Net Work Done in a Cycle

The net work done in a thermodynamic cycle is obtained by summing the work done during each segment of the cycle. In a PV diagram, this net work is represented by the area enclosed by the cycle. It is a key metric in determining the performance and efficiency of a heat engine.

Thermal Efficiency

Thermal efficiency is the measure of a heat engine’s ability to convert absorbed heat into useful work. It is defined as the ratio of net work output to the heat input during the cycle. Calculating thermal efficiency involves determining both the total heat absorbed and the resultant work done, providing insight into the engine’s performance.

First Law of Thermodynamics

The first law of thermodynamics, which is a statement of energy conservation, asserts that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system (?U = Q ? W). In cyclic processes, because the system ends where it began, the change in internal energy is zero, meaning that the net work done over the cycle is exactly equal to the net heat exchange.

Transcript

00:01 Hi everyone here it is given a heat engine that takes one mole of ideal diatomic gas through the cycle as shown in the figure v to c c to a and a to b.

00:29 So i am so right a here in the first part we have to justify the av is isothermal compression v part during which segment the cycle is absorbed in which segment q is absorbed c part calculate temperature of a, b and c point v part we have to calculate net heat exchange with surrounding q exchange with surrounding and work done and we have to calculate thermal efficiency of the cycle.

02:32 Let us start solving it.

02:34 First part, we'll start with part a using guess equation we can find p .a.

03:04 Into va that is 2 .0 into 10 to the power 3 joules.

03:12 Product of pressure and volume at v point and using pv is called to nrt we will find since pavav is called to pv pv so temperature of a is called to temperature of v so av is isothermal compression this is the answer of a part.

04:21 Now we will start b for isothermal amount of the heat exchange is called to never done in the cycle that is nrt loan of v2 upon v1 since a .b is compression.

05:09 So q is greater than zero that is heat is absorbed.

05:26 Q is called in since a veh compression with just a moment let me correct.

05:41 With volume of b is less than a so q is less than 0 that is heat is rejected bc is at constant pressure so q is called to n cp delta t or cp by r p delta v here delta v is greater than zero so q is greater than zero that is heat is absorbed cd is at constant volume q is called to n cv delta t and it can be written as cv upon r v delta p here delta p is less than zero hence heat is rejected so this is the answer of part a.

08:23 I'm underlining in a .b process heat is rejected...

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