Learning Goal: To derive the expression for the work done by an expanding gas, d

Question

Learning Goal:

To derive the expression for the work done by an expanding gas, dW=pdV, and to understand how it follows from the expression W=Fd for mechanical work.

Especially from the historically important perspective of making engines to convert heat energy into work, the work in thermodynamics is defined as the work done by the system on the exterior world, and not vice versa as is done in the rest of classical mechanics. In classical mechanics, one always considers the work done on a system by the outside world. Rarely does one think about the work done by the system. Suppose you push a large block with a certain force of magnitude F over some distance. You have done work on the block; hence the energy of the block should increase. According to Newton's 3rd law, the block exerts the same magnitude of force F, but in the opposite direction (i.e., directed back at you). Hence, the work done by the block (on you) is negative, since the direction of motion opposes the direction of the force. In summary, you have to be careful about the sign of the work: the same situation gives opposite signs of the work depending on whether our perspective is classical mechanics or thermodynamics.

In thermodynamics, one often deals with liquids and gases that exert forces on their containers (i.e., the fluids exert pressure over an area). If the container changes volume, then this force acts through a distance and hence does work.

For a steam engine, the example pictured here,(Figure 1) the "container" is a cylinder whose volume changes as the piston slides in or out. Suppose a gas is confined within the cylinder. The pressure of the gas is p, and the area of the cylinder is A. Consider the work done as the gas expands, pushing the piston to the right. Call the infinitesimal distance the piston moves dx.

Part A

What force F does the gas exert on the piston? (Note that the positive x axis is to the right in the figure.)

Express the force in terms of p, A, and any constants,

SubmitMy AnswersGive Up

Part B

If the piston moves a distance dx, what is dW, the work done by the gas?

Express the work done by the gas in terms of given quantities.

pdV

SubmitHintsMy AnswersGive UpReview Part

Incorrect; Try Again; 4 attempts remaining

Part C

What is dV, the increase in volume of the gas?

Express the differential increase in terms of dx and other given quantities.

SubmitHintsMy AnswersGive UpReview Part

Part D

This question will be shown after you complete previous question(s).

Part E

Suppose that the gas expands from V0 to V1 at constant pressure p0. How much work W is done by the gas?

Express the work in terms of p0, V0, and V1.

SubmitMy AnswersGive Up

Part F

This question will be shown after you complete previous question(s).

Part G

Assume now that the diameter of the piston is reduced by a factor of 2. What is the amount of work W2D done by a gas of pressure p0 in expanding from the same initial volume V0 to the same final volume V1? Note that the piston has to start out much farther to the right for the volume to be V0 initially.

Express your answer in terms of p0, V0, V1, and simple numerical factors.

SubmitMy AnswersGive Up

Provide FeedbackContinue

Figure 1 of 1

Part A

What force F does the gas exert on the piston? (Note that the positive x axis is to the right in the figure.)

Express the force in terms of p, A, and any constants,

F =

SubmitMy AnswersGive Up

Part B

If the piston moves a distance dx, what is dW, the work done by the gas?

Express the work done by the gas in terms of given quantities.

dW =

pdV

SubmitHintsMy AnswersGive UpReview Part

Incorrect; Try Again; 4 attempts remaining

Part C

What is dV, the increase in volume of the gas?

Express the differential increase in terms of dx and other given quantities.

dV =

SubmitHintsMy AnswersGive UpReview Part

Part D

This question will be shown after you complete previous question(s).

Part E

Suppose that the gas expands from V0 to V1 at constant pressure p0. How much work W is done by the gas?

Express the work in terms of p0, V0, and V1.

W =

SubmitMy AnswersGive Up

Part F

This question will be shown after you complete previous question(s).

Part G

Assume now that the diameter of the piston is reduced by a factor of 2. What is the amount of work W2D done by a gas of pressure p0 in expanding from the same initial volume V0 to the same final volume V1? Note that the piston has to start out much farther to the right for the volume to be V0 initially.

Express your answer in terms of p0, V0, V1, and simple numerical factors.

W2D =

SubmitMy AnswersGive Up

Provide FeedbackContinue

Figure 1 of 1

Explanation / Answer

part A: Pressure = force/area

so Force = PA

------------------------------------------------------

part B : Amount of work done W = pdV = F dx

so W = P A dx
------------------------------------------------------

part C: since volume V = area * displacement

dV = A * dx

--------------------------------------------------------------

PART E), from aboice we have increase in volume, dV = V1 - V0

And pressure is given as , P = 0

Work done is given as, W = PdV = P0.(V1 - V0)

------------------------------------------------------------------

PART G)

There is no effect of diameter on the work done as it only depends of final and initial volume of the gas

Here initial volume and final volume is same as well as external pressure is also same

So work done is given as, W2D = P0.(V1 - V0)
-------------------------------------------------------------------------------------

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.