When gas expands in a cylinder with radius r , the pressure P at any given time

Question

When gas expands in a cylinder with radius r, the pressure P at any given time is a function of the volume V: P = P(V). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F = r2P. The work done by the gas when the volume expands from volume V1 to volume V2 is

W =

.

In a steam engine the pressure and volume of steam satisfy the equation PV1.4 = k, where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Calculate the work done by the engine during a cycle when the steam starts at a pressure of 160 lb/in2 and a volume of 100 in3 and expands to a volume of 600 in3. (Round your answer to two decimal places.)

Explanation / Answer

Here is the given data

160 lb/in^2 = (160*144) lb/ft^2
100 in^3 = 100/1728 ft^3 = V1
600 in^3 = 600/1728 ft^3 = V2

We have to find P(V), remember since PV^(1.4) = k

k = (160*144)(100/1728)^(1.4) =426.50

now we will find work by using integral

W = integral from V1 to V2 of P(V) dV

so integral from (100/1728) to (600/1728) of 426.50*V^(-1.4) dV

0.06 to0.35 of   426.50 V^(-1.4) dV

=3.90 answer

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