Solution for Question 16
Analysis:
Both balloons start at the same depth (same Pressure $P_1$) and temperature (Lake Temperature $T$). As they rise, the surrounding pressure decreases to $P_2$. The gas in both balloons expands.
1. First Balloon (Conducting):
The rubber is a good conductor. As the gas expands, it tends to cool down, but heat flows in rapidly from the lake water (uniform temperature). The process is Isothermal ($T = \text{constant}$).
$$P_1 V_1 = P_2 V_{iso} \Rightarrow V_{iso} = V_1 \left( \frac{P_1}{P_2} \right)$$
2. Second Balloon (Insulating):
The rubber is an insulator. Heat cannot flow in from the lake. The expansion is Adiabatic.
$$P_1 V_1^\gamma = P_2 V_{adia}^\gamma \Rightarrow V_{adia} = V_1 \left( \frac{P_1}{P_2} \right)^{1/\gamma}$$
Comparison:
Since the balloons rise, $P_1 > P_2$, so the ratio $\frac{P_1}{P_2} > 1$.
For any gas, $\gamma > 1$ (e.g., 1.4 for air), so the exponent $\frac{1}{\gamma} < 1$.
Mathematically, for a base $X > 1$, $X^1 > X^{fraction}$. Therefore:
$$V_{iso} > V_{adia}$$
The conducting balloon (isothermal expansion) expands more because it absorbs heat energy to maintain its temperature, whereas the insulating balloon uses its own internal energy to do the work of expansion, causing it to cool and expand less.