One of the two vessels of same capacity is filled with oxygen and other is filled with helium of same mass. Both gasess are at same temperature. The ratio of pressure of these gases is

Options

A. `(P_(0))/(P_(He))=1/8`
B. `(P_(0))/(P_(He))=1/4`
C. `(P_(0))/(P_(He))=4/1`
D. `(P_(0))/(P_(He))=8/1`

Explanation

To find the ratio of the pressures of oxygen and helium gases in two vessels of the same capacity, filled with the same mass of gas at the same temperature, we can use the ideal gas law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant, and
- \( T \) is the temperature.

### Step-by-Step Solution:

1. **Write the Ideal Gas Law for Both Gases**:
For oxygen (O₂):
\[ P_{O_2} V = n_{O_2} R T \]
For helium (He):
\[ P_{He} V = n_{He} R T \]

2. **Take the Ratio of Pressures**:
Dividing the equations for both gases:
\[ \frac{P_{O_2}}{P_{He}} = \frac{n_{O_2} R T}{n_{He} R T} \]
Here, \( R \) and \( T \) are constants and will cancel out:
\[ \frac{P_{O_2}}{P_{He}} = \frac{n_{O_2}}{n_{He}} \]

3. **Relate Moles to Mass**:
The number of moles \( n \) can be expressed in terms of mass \( m \) and molar mass \( M \):
\[ n = \frac{m}{M} \]
Thus, we can write:
\[ n_{O_2} = \frac{m}{M_{O_2}} \quad \text{and} \quad n_{He} = \frac{m}{M_{He}} \]

4. **Substitute into the Pressure Ratio**:
Substituting the expressions for \( n_{O_2} \) and \( n_{He} \) into the pressure ratio:
\[ \frac{P_{O_2}}{P_{He}} = \frac{\frac{m}{M_{O_2}}}{\frac{m}{M_{He}}} \]
The mass \( m \) cancels out:
\[ \frac{P_{O_2}}{P_{He}} = \frac{M_{He}}{M_{O_2}} \]

5. **Insert Molar Mass Values**:
The molar mass of helium \( M_{He} \) is 4 g/mol, and the molar mass of oxygen \( M_{O_2} \) is 32 g/mol:
\[ \frac{P_{O_2}}{P_{He}} = \frac{4}{32} = \frac{1}{8} \]

6. **Conclusion**:
Therefore, the ratio of the pressures of oxygen to helium is:
\[ \frac{P_{O_2}}{P_{He}} = \frac{1}{8} \]

### Final Answer:
The ratio of the pressure of oxygen to the pressure of helium is \( \frac{1}{8} \).

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