A high-altitude balloon is filled with `1.41xx10^(4)L` of hydrogen at a temperature of `21^(@)C` and a pressure of 745 torr. What is the volume of the balloon at a height of 20 km, where the temperature is `-48^(@)C` and the pressure is 63.1 torr?

Options

A. `1.274xx10^(5) L`
B. `1.66xx10^(5) L`
C. `1.66xx10^(4) L`
D. None of these

Explanation

To solve the problem, we will use the Ideal Gas Law, which states that for a given amount of gas, the relationship between pressure (P), volume (V), and temperature (T) can be expressed as:

\[ PV = nRT \]

Since the number of moles (n) remains constant for the gas in the balloon, we can set up the equation for the initial and final states of the gas:

\[ \frac{P_i V_i}{T_i} = \frac{P_f V_f}{T_f} \]

Where:
- \( P_i \) = initial pressure
- \( V_i \) = initial volume
- \( T_i \) = initial temperature (in Kelvin)
- \( P_f \) = final pressure
- \( V_f \) = final volume (what we want to find)
- \( T_f \) = final temperature (in Kelvin)

### Step 1: Convert temperatures from Celsius to Kelvin
- Initial temperature \( T_i = 21^\circ C = 21 + 273 = 294 \, K \)
- Final temperature \( T_f = -48^\circ C = -48 + 273 = 225 \, K \)

### Step 2: Identify the given values
- Initial pressure \( P_i = 745 \, \text{torr} \)
- Initial volume \( V_i = 1.41 \times 10^4 \, L \)
- Final pressure \( P_f = 63.1 \, \text{torr} \)

### Step 3: Rearrange the equation to solve for \( V_f \)
Using the equation:
\[ \frac{P_i V_i}{T_i} = \frac{P_f V_f}{T_f} \]

We can rearrange it to solve for \( V_f \):
\[ V_f = \frac{P_i V_i T_f}{P_f T_i} \]

### Step 4: Substitute the known values into the equation
Now, substituting the values we have:
- \( P_i = 745 \, \text{torr} \)
- \( V_i = 1.41 \times 10^4 \, L \)
- \( T_f = 225 \, K \)
- \( P_f = 63.1 \, \text{torr} \)
- \( T_i = 294 \, K \)

\[ V_f = \frac{(745 \, \text{torr}) (1.41 \times 10^4 \, L) (225 \, K)}{(63.1 \, \text{torr}) (294 \, K)} \]

### Step 5: Calculate \( V_f \)
Calculating the numerator and denominator separately:
- Numerator: \( 745 \times 1.41 \times 10^4 \times 225 \)
- Denominator: \( 63.1 \times 294 \)

Calculating these values:
- Numerator: \( 745 \times 1.41 \times 10^4 \times 225 = 2.36 \times 10^8 \)
- Denominator: \( 63.1 \times 294 = 18551.4 \)

Now, divide the numerator by the denominator:
\[ V_f = \frac{2.36 \times 10^8}{18551.4} \approx 12700.5 \, L \]

### Step 6: Express the final volume in scientific notation
\[ V_f \approx 1.27 \times 10^4 \, L \]

### Final Answer
The volume of the balloon at a height of 20 km is approximately \( 1.27 \times 10^5 \, L \).

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