Question

A gas in a piston-cylinder asscmbly undcrgocs a process for which the rclationship bctwcen pressurc and volumc is pV^2=constant The initial pressurc is 1 bar, the initial volume is 0.4 m^3, and the final pressure is 9 bar. Determine the work for the process, in kJ constant.

Answer 1

Given:

$pV^{2}$ = constant                                   (1)

⇒ $p_{1}V_{1}^{2} = p_{2}V_{2}^{2}$          (2)

$p_{1} = 1 bar = 1\times 10^{5}$

$p_{2} = 9 bar = 9\times 10^{5}$

$V_{1} = 0.4 m^{3}$

$V_{2} = ? m^{3}$

Solution:

Here, from eqn (1),  the polytropic constant is '2' ( Since, here $pV^{n}$ = $pV^{2}$ )

Now, using eqn (2), we get

$V_{2}^{2} =\frac{p_{2}}{p_{1}}\times V_{1}^{2}$

putting the values in above eqn, we get-

$V_{2}^{2} =\frac{9}{1}\times 0.4^{2}$

$V_{2} = 1.2 m^{3}$

Now, work for the process is given by:

$W = \frac{p_{2}V_{2} - p_{1}V_{1}}{1 - n}$                  (3)

where,

n = potropic constant = 2

Using Eqn (3), we get:

$W = \frac{9\times 10^{5}\times 1.2 - 1\times 10^{5}\times 0.4}{1 - 2}$

W = - 240 kJ