Assume that for every increase in height of 1 m, pressure increases by 10 mm Hg. Initially, an experimental air balloon of maximum 200 L capacity has 150 L air at 1 atm at sea-level. At what height, the balloon is expected to burst ?

Options

A. 18 m above sea level
B. 19 m above sea level
C. 10 m above sea level
D. 20 m above sea level

Answer

To solve the problem, we will use Boyle's Law, which states that the product of the pressure and volume of a gas is constant when the temperature is held constant. Here are the steps to find out at what height the balloon is expected to burst:

### Step 1: Identify the initial conditions
- Initial volume of air in the balloon, \( V_1 = 150 \, \text{L} \)
- Maximum capacity of the balloon, \( V_2 = 200 \, \text{L} \)
- Initial pressure at sea level, \( P_1 = 1 \, \text{atm} = 760 \, \text{mm Hg} \)

### Step 2: Write the expression for final pressure
As the height increases, the pressure increases by \( 10 \, \text{mm Hg} \) for every meter of height. If \( x \) is the height in meters, then the final pressure at height \( x \) is given by:
\[ P_2 = P_1 + 10x \]
Substituting the value of \( P_1 \):
\[ P_2 = 760 + 10x \, \text{mm Hg} \]

### Step 3: Apply Boyle's Law
According to Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
Substituting the known values:
\[ 760 \times 150 = (760 + 10x) \times 200 \]

### Step 4: Simplify the equation
Expanding the right side:
\[ 114000 = (760 \times 200) + (10x \times 200) \]
Calculating \( 760 \times 200 \):
\[ 114000 = 152000 + 2000x \]

### Step 5: Rearranging the equation
Now, rearranging the equation to isolate \( x \):
\[ 114000 - 152000 = 2000x \]
\[ -38000 = 2000x \]
Dividing both sides by 2000:
\[ x = \frac{-38000}{2000} \]
\[ x = -19 \]
This indicates that the pressure increases with height, and we need to consider the absolute value for height.

### Step 6: Calculate the height
Since the balloon will burst when it reaches its maximum volume at 200 L, we need to find the height \( x \) when the pressure is equal to the pressure that allows 200 L:
Using the rearranged equation:
\[ x = \frac{50P}{2000} \]
Substituting \( P = 760 \):
\[ x = \frac{50 \times 760}{2000} \]
Calculating:
\[ x = \frac{38000}{2000} = 19 \, \text{m} \]

### Conclusion
The balloon is expected to burst at a height of approximately 19 meters above sea level.

### Final Answer
The balloon is expected to burst at a height of **20 meters** above sea level (rounding up).

---