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. Calculate : (a) No. of moles of air added to change the pressure from 1 atm to 3 atm. (b) balloon will burst if either pressure increases to 7 atm or volume increases to 36π m³. Calculate the number of moles of air that must be added after initial condition to burst the balloon

• A. (a) 80 moles (b) 1295 moles
• B. (a) 81 moles (b) 2401 moles
• C. (a) 80 moles (b) 2400 moles
• D. (a) 81 moles (b) 1296 moles

Answer 1

Answer: (A)

(a) 80 moles (b) 1295 moles

Answer 2

Assume constant temperature $T$. Given $P \propto d$, let $P=kd$. With $P_1=1$ atm, $d_1=1$ m, we get $k=1$ atm/m, so $P=d$. Volume of balloon $V = \frac{\pi d^3}{6}$. From Ideal Gas Law $PV=nRT$, we have $\frac{PV}{n} = \text{constant}$. Substituting $P=kd$ and $V=\frac{\pi d^3}{6}$: $\frac{(kd)(\frac{\pi d^3}{6})}{n} = \text{constant} \implies \frac{d^4}{n} = \text{constant}$. Thus, $n = n_1 \left(\frac{d}{d_1}\right)^4$.

(a) For $P_2=3$ atm, since $P=d$, $d_2=3$ m. $n_2 = n_1 \left(\frac{d_2}{d_1}\right)^4 = 1 \text{ mol} \times \left(\frac{3 \text{ m}}{1 \text{ m}}\right)^4 = 1 \times 3^4 = 81$ moles. Moles added = $n_2 - n_1 = 81 - 1 = 80$ moles.

(b) Balloon bursts if $P=7$ atm OR $V=36\pi$ m³. Condition 1: $P_{burst1} = 7$ atm $\implies d_{burst1}=7$ m. $n_{burst1} = n_1 \left(\frac{d_{burst1}}{d_1}\right)^4 = 1 \text{ mol} \times \left(\frac{7 \text{ m}}{1 \text{ m}}\right)^4 = 1 \times 7^4 = 2401$ moles. Moles added = $2401 - 1 = 2400$ moles.

Condition 2: $V_{burst2} = 36\pi$ m³. $36\pi = \frac{\pi d_{burst2}^3}{6} \implies d_{burst2}^3 = 216 \implies d_{burst2} = 6$ m. $n_{burst2} = n_1 \left(\frac{d_{burst2}}{d_1}\right)^4 = 1 \text{ mol} \times \left(\frac{6 \text{ m}}{1 \text{ m}}\right)^4 = 1 \times 6^4 = 1296$ moles. Moles added = $1296 - 1 = 1295$ moles.

The balloon bursts when the minimum number of moles are added to reach either condition. Minimum moles added = min(2400, 1295) = 1295 moles.