A weather balloon is inflated with helium. The balloon has a volume of 100 `m^(3)` and it must be inflated to pressure of 0.10 atm. If 50 L gas cylinders of helium at a pressure of 100 atm are used, how many cylinders are needed? Assume that the temperature is constant.

Options

A. 2
B. 3
C. 4
D. 1

Explanation

To solve the problem, we will use the ideal gas law and the relationship between pressure, volume, and the number of cylinders needed.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Volume of the weather balloon (V_balloon) = 100 m³
- Pressure of the balloon (P_balloon) = 0.10 atm
- Volume of one gas cylinder (V_cylinder) = 50 L
- Pressure of one gas cylinder (P_cylinder) = 100 atm

2. **Convert the Volume of the Cylinder to Cubic Meters**:
- Since 1 m³ = 1000 L, we can convert the volume of the cylinder from liters to cubic meters.
\[ V_cylinder = 50 \, \text{L} = \frac{50}{1000} \, \text{m}^3 = 0.05 \, \text{m}^3 \]

3. **Calculate the Volume of Helium Available from One Cylinder at Balloon Pressure**:
- Using the ideal gas law, we can find out how much volume of gas at the balloon's pressure can be obtained from one cylinder at its pressure.
- According to the ideal gas law, \( P_1V_1 = P_2V_2 \), where:
- \( P_1 \) = Pressure of the cylinder = 100 atm
- \( V_1 \) = Volume of the cylinder = 0.05 m³
- \( P_2 \) = Pressure of the balloon = 0.10 atm
- \( V_2 \) = Volume of gas at balloon pressure (what we want to find)

Rearranging the equation gives:
\[ V_2 = \frac{P_1 \cdot V_1}{P_2} = \frac{100 \, \text{atm} \cdot 0.05 \, \text{m}^3}{0.10 \, \text{atm}} = \frac{5}{0.1} = 50 \, \text{m}^3 \]

4. **Determine How Many Cylinders are Needed**:
- Now we know that one cylinder can provide 50 m³ of helium at the balloon's pressure.
- To find out how many cylinders are needed for 100 m³:
\[ \text{Number of cylinders} = \frac{V_{balloon}}{V_2} = \frac{100 \, \text{m}^3}{50 \, \text{m}^3} = 2 \]

### Final Answer:
You will need **2 cylinders** of helium to inflate the weather balloon.

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