1 litre of `N_(2) " and " 7//8` litre of `O_(2)` at the same temperature and pressure were mixed together. What is the relation between the masses of the two gases in the mixture ?

Options

A. `M_(N_(2)) = 3M_(O_(2))`
B. `M_(N_(2)) = 8M_(O_(2))`
C. `M_(N_(2)) = M_(O_(2))`
D. `M_(N_(2)) =16 M_(O_(2))`

Answer

To solve the problem of finding the relation between the masses of nitrogen gas (N₂) and oxygen gas (O₂) when 1 liter of N₂ is mixed with 7/8 liter of O₂ at the same temperature and pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law states that \( PV = nRT \), where \( n \) is the number of moles, \( R \) is the gas constant, \( T \) is temperature, and \( P \) is pressure. We can express the number of moles in terms of mass and molar mass:
\[ n = \frac{m}{M} \]
where \( m \) is mass and \( M \) is molar mass.

2. **Write the equations for both gases**:
For nitrogen (N₂):
\[ P \cdot V_{N_2} = \frac{m_{N_2}}{M_{N_2}} \cdot RT \]
Given \( V_{N_2} = 1 \, \text{L} \) and \( M_{N_2} = 28 \, \text{g/mol} \):
\[ P \cdot 1 = \frac{m_{N_2}}{28} \cdot RT \quad \text{(Equation 1)} \]

For oxygen (O₂):
\[ P \cdot V_{O_2} = \frac{m_{O_2}}{M_{O_2}} \cdot RT \]
Given \( V_{O_2} = \frac{7}{8} \, \text{L} \) and \( M_{O_2} = 32 \, \text{g/mol} \):
\[ P \cdot \frac{7}{8} = \frac{m_{O_2}}{32} \cdot RT \quad \text{(Equation 2)} \]

3. **Set up the equations**:
From Equation 1:
\[ P = \frac{m_{N_2}}{28} \cdot \frac{RT}{1} \]

From Equation 2:
\[ P = \frac{m_{O_2}}{32} \cdot \frac{RT}{\frac{7}{8}} \]

4. **Equate the two expressions for pressure (P)**:
\[ \frac{m_{N_2}}{28} = \frac{m_{O_2}}{32} \cdot \frac{8}{7} \]

5. **Cross-multiply to solve for the relation between masses**:
\[ 7 \cdot m_{N_2} \cdot 32 = 8 \cdot m_{O_2} \cdot 28 \]
Simplifying gives:
\[ 224 \cdot m_{N_2} = 224 \cdot m_{O_2} \]

6. **Conclude the relation**:
\[ m_{N_2} = m_{O_2} \]
Thus, the masses of nitrogen and oxygen in the mixture are equal.

### Final Answer:
The relation between the masses of \( N_2 \) and \( O_2 \) in the mixture is:
\[ m_{N_2} = m_{O_2} \]