Question

A balloon behaves so that the pressure isP=C2V1/3 and C2 = 100 kPa/m. The balloon is blown up with air from a starting volume of 1 m3 to a volume of 4 m3. Find the final mass of air, assuming it is at 25◦C, and the work done by the air.

Answer 1

Step-by-step explanation:

As it is given that,

`P_(1) = C_(2)V^{(1)/(3)}`

`P_(1)V^{-(1)/(3)}_(1) = C_(2)`

As the system is not gaining or losing heat. So, it is an adiabatic process in an assumed ideal gas. The polytropic extent n is
`(-1)/(3)`.

`P_(1) = C_(1)V(1)/(3)_(1)`

=
`(100)(1)^{(1)/(3)}`

= 100 kpa

`P_(2) = C_(2)V^{(1)/(3)}_(2)`

=
`100(4)^{(1)/(3)}`

= 158.74 kpa

Now, work done by the air is as follows.

`W_(2) = \int PdV`

=
`(P_(2)V_(2) - P_(1)V_(1))/(1 - n)`

=
`(158.74 * 4 - 100 * 1)/(1 - ((-1)/(3)))`

= 401.22 kJ

Work done by the air is as follows.

`P_(2)V_(2) = m_(2)RT_(2)`

`m_(2) = (P_(2)V_(2))/(RT_(2))`

=
`(158.74 * 4)/(0.287 * (273 + 25))`

= 7.424 kg

Thus, we can conclude that final mass of air is 7.424 kg and work done by the air is 401.22 kJ.