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

Explanation:

As it is given that,

            $P_{1} = C_{2}V^{\frac{1}{3}}$

           $P_{1}V^{-\frac{1}{3}}_{1} = C_{2}$

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 $\frac{-1}{3}$.

            $P_{1} = C_{1}V\frac{1}{3}_{1}$

                       = $(100)(1)^{\frac{1}{3}}$

                        = 100 kpa

         $P_{2} = C_{2}V^{\frac{1}{3}}_{2}$

                      = $100(4)^{\frac{1}{3}}$

                      = 158.74 kpa

Now, work done by the air is as follows.

            $W_{2} = \int PdV$

                       = $\frac{P_{2}V_{2} - P_{1}V_{1}}{1 - n}$

                       = $\frac{158.74 \times 4 - 100 \times 1}{1 - (\frac{-1}{3})}$

                       = 401.22 kJ

Work done by the air is as follows.

           $P_{2}V_{2} = m_{2}RT_{2}$

               $m_{2} = \frac{P_{2}V_{2}}{RT_{2}}$

                           = $\frac{158.74 \times 4}{0.287 \times (273 + 25)}$

                           = 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.