Chapter 15.9, Problem 22E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

An important problem in thermodynamics is to find the work done by an ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is equal to the area of the region *R* enclosed by two isothermal curves *xy*= *a*. *xy = b* and two adiabatic curves *xy*^{1.4} = *c*, *xy*^{1.4} = *d*, where 0 < *a < b* and 0 *< c < d.* Compute the work done by determining the area of *R.*

To determine

**To compute:** The work done by the engine.

Explanation

**Given:**

The region *R* enclosed by the isothermal curves are

The adiabatic curves are,

The region varies

**Calculation:**

The region *R* is defined as

Choose the given curves as

Then, the region *R* becomes,

Let

Obtain the value of *y* by substituting
*v*.

Raise to the power

Substitute
*u*,

Obtain the Jacobian,

Find the partial derivative of *x* and *y* with respect to *u* and *v.*

If

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