Question

A balloon containing $1$ mole air at $1$ atm initially is filled further with air till pressure increases to $3$ atm. The initial diameter of the balloon is $1$ m and the pressure at each state is proportion to diameter of the balloon. Balloon will burst if either pressure increases to $7$ atm or volume increases to $36\pi m^3$. Calculate the number of moles of air that must be added after initial condition to burst the balloon.

Answer 1

Initial no. of moles=$n_1$=1 mole

Initial Pressure=$P_1$=1 atm

Initial diameter=$d_1$=1 m

Volume of the balloon=$\frac{4}{3}\pi r^3=\frac{4}{3}\pi {\left(\frac{d}{2}\right)}^3=\frac{4}{3}\pi {\left(\frac{1}{2}\right)}^3$

As pressure is directly proportional to the diameter. Thus,

Final Pressure=$P_2$=3 atm

Final diameter=$d_2$=3 m

Volume of the balloon=$V_2=\frac{4}{3}\pi r^3=\frac{4}{3}\pi {\left(\frac{d}{2}\right)}^3=\frac{4}{3}\pi {\left(\frac{3}{2}\right)}^3$

Now, by using the ideal gas equation,

$\frac{P_1{V}_1}{n_1}=\frac{P_2{V}_2}{n_2}⟹\frac{1atm\times \frac{4}{3}\pi {\left(\frac{1}{2}\right)}^3}{1}=\frac{3atm\times \frac{4}{3}\pi {\left(\frac{3}{2}\right)}^3}{n_2}⟹n_2=3\times 3^3=81$

Thus, 81 moles present finally.

So, no. of moles added to change the pressure is=81-1=80 moles.