Sketch a P V diagram and find the work done by the gas...

Sketch a $P V$ diagram and find the work done by the gas during the following stages. (a) A gas is expanded from a volume of 1.0 $\mathrm{L}$ to 3.0 $\mathrm{L}$ at a constant pressure of 3.0 $\mathrm{atm} .$ (b) The gas is then cooled at constant volume until the pressure falls to 2.0 atm. (c) The gas is then compressed at a constant pressure of 2.0 atm from a volume of 3.0 $\mathrm{L}$ to 1.0 $\mathrm{L}$ . Note: Be careful of signs. (d) The gas is heated until its pressure increases from 2.0 atm to 3.0 $\mathrm{atm}$ at a constant volume. (e) Find the net work done during the complete cycle.

Key Concepts

Sign Convention in Work

Correctly applying the sign convention for work is crucial in thermodynamics. Typically, work done by the gas on the surroundings (such as during expansion) is taken as positive, while work done on the gas (such as during compression) is negative. This convention helps in correctly analyzing energy transfers and maintaining consistency across various processes.

Cyclic Process

A cyclic process is one in which the system returns to its initial state after undergoing a series of thermodynamic changes. The net work done over a complete cycle is equal to the area enclosed by the cycle on the PV diagram. This concept is fundamental in the study of heat engines and refrigerators, where the net work represents the efficiency of the cycle.

Constant Volume Process

During a constant volume process, the volume of the gas remains unchanged even if other parameters such as pressure vary. As a result, no work is done by or on the gas because work in these processes depends on a change in volume. On a PV diagram, this process is represented by a vertical line.

Work Done by a Gas

The work done by a gas during a thermodynamic process is defined as the integral of pressure with respect to volume. This value represents the energy transferred when the gas expands or compresses and is calculated by integrating the pressure over the change in volume (W = ?P dV). For constant pressure processes, this simplifies to the product of pressure and volume change.

Pressure-Volume (PV) Diagram

A PV diagram is a graphical representation of the state of a thermodynamic system, plotting pressure on the vertical axis and volume on the horizontal axis. It visually illustrates various thermodynamic processes and the relationships between pressure, volume, and work, with the area under or enclosed by the process path corresponding to the work done by or on the system.

Constant Pressure Process

In a constant pressure process, the pressure remains fixed while the volume changes. The simplicity of this process allows the work done to be computed directly by multiplying the constant pressure by the change in volume. This type of process is common in many practical applications and is easily represented on a PV diagram as a horizontal line.

Transcript

0:00 Here we have a situation.

00:03 We have here on x -axis a volume and on y -axis we have a pressure.

00:11 So i may label the volume.

00:13 This is volume.

00:15 One, two, three.

00:18 This is in atm for now.

00:22 And this pressure, pressure, sorry, this is in liters.

00:26 And pressure here is in atm.

00:29 So atm.

00:29 So, atm.

00:31 So this is.

00:32 This is 1 litre, 2, 3 liter, this is 1, 2, 2, and 3 atm, sorry.

00:40 So the gase is expanded, so expanded from volume 1 to 3, and then it is at a constant pressure, and then it is at constant volume is compressed, and then it is compressed volume, so here we have a volume constant.

01:08 So let me call this one, this is two, and this is three.

01:12 And then at a constant volume it is heated, so the pressure increases to the three again back.

01:19 So here we have a pressure two, here three.

01:22 So this segment is actually called to level four.

01:28 Then the work done in phase one, so work done in this phase, in phase one is simply the pressure times pressure times del v so pressure we have here is three three atm so we can convert three atms into the pascals so which will be 3 .03 times 10 to the power 5 pascals the change in volume we see is a two two liter so we converting in two liters in a meter cube we have 2 times 10 to the power minus 3 meter cube...

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