Inside a spherical glass flask 'A' of radius 1 meter containg 300gm of H2, there was another rubber balloon B containing some N2. Inside the balloon B, another rubber balloon 'C containing some `O_(2)` is present. At 27°C it was found that the balloon B had a radius of 60cm and balloon 'C had a radius of 30cm
Calculate the moles of nitrogen in the balloon B

Options

A. 4.123atm
B. 3.123 atm
C. 2.123 atm
D. 1.123 atm

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the volume of balloon B and balloon C
- The radius of balloon B (r_B) = 60 cm = 0.6 m
- The radius of balloon C (r_C) = 30 cm = 0.3 m

Using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]

**Volume of balloon B (V_B):**
\[ V_B = \frac{4}{3} \pi (0.6)^3 \]
\[ V_B = \frac{4}{3} \pi (0.216) \]
\[ V_B \approx 0.904 \, \text{m}^3 \]

**Volume of balloon C (V_C):**
\[ V_C = \frac{4}{3} \pi (0.3)^3 \]
\[ V_C = \frac{4}{3} \pi (0.027) \]
\[ V_C \approx 0.113 \, \text{m}^3 \]

### Step 2: Calculate the volume of gas in balloon B
The volume of gas in balloon B is the volume of balloon B minus the volume of balloon C:
\[ V_{gas} = V_B - V_C \]
\[ V_{gas} = 0.904 - 0.113 \]
\[ V_{gas} \approx 0.791 \, \text{m}^3 \]

### Step 3: Calculate the moles of H2 in the flask
The mass of H2 given is 300 g. To find the number of moles (n), we use the formula:
\[ n = \frac{mass}{molar \, mass} \]
The molar mass of H2 = 2 g/mol:
\[ n_{H2} = \frac{300 \, g}{2 \, g/mol} = 150 \, mol \]

### Step 4: Calculate the pressure of H2 in the flask
Using the ideal gas law:
\[ PV = nRT \]
Where:
- P = pressure
- V = volume of the gas
- n = number of moles
- R = ideal gas constant (0.0821 L·atm/(K·mol))
- T = temperature in Kelvin (27°C = 300 K)

The volume of the flask A:
\[ V_A = \frac{4}{3} \pi (1)^3 \approx 4.18879 \, m^3 \]

Now, substituting the values into the ideal gas law:
\[ P_{H2} = \frac{nRT}{V_A} \]
\[ P_{H2} = \frac{150 \times 0.0821 \times 300}{4.18879} \]
Calculating this gives:
\[ P_{H2} \approx 1.176 \, atm \]

### Step 5: Calculate the moles of nitrogen in balloon B
Using the ideal gas law for balloon B:
\[ n_{N2} = \frac{PV_{gas}}{RT} \]
Where:
- P = P_H2 (pressure of H2 in balloon B)
- V = V_gas (volume of gas in balloon B)
- R = 0.0821 L·atm/(K·mol)
- T = 300 K

Substituting the values:
\[ n_{N2} = \frac{1.176 \times 0.791}{0.0821 \times 300} \]
Calculating this gives:
\[ n_{N2} \approx 0.031 \, mol \]

### Final Answer
The moles of nitrogen in balloon B is approximately **0.031 mol**.

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