The ratio of the densities of hydrogen and oxygen at S.T.P is

Options

A. `4 : 1`
B. `1 : 16`
C. `1 : 4`
D. `16 : 1`

Answer

To find the ratio of the densities of hydrogen and oxygen at standard temperature and pressure (S.T.P), we can follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 2: Express Density in Terms of the Ideal Gas Law
The density (\( D \)) of a gas can be expressed as:
\[ D = \frac{mass}{volume} \]
We know that the number of moles (\( n \)) can be expressed as:
\[ n = \frac{W}{M} \]
where:
- \( W \) = weight of the gas
- \( M \) = molecular weight of the gas

Substituting this into the ideal gas law gives:
\[ PV = \frac{W}{M} RT \]
Rearranging for density, we have:
\[ D = \frac{W}{V} = \frac{PM}{RT} \]

### Step 3: Calculate the Densities of Hydrogen and Oxygen
At S.T.P (Standard Temperature and Pressure):
- Pressure (\( P \)) = 1 atm = 101.325 kPa
- Temperature (\( T \)) = 273 K

The molecular weights are:
- Molecular weight of Hydrogen (\( H_2 \)) = 2 g/mol (1 g/mol for each hydrogen atom)
- Molecular weight of Oxygen (\( O_2 \)) = 32 g/mol (16 g/mol for each oxygen atom, and there are 2 oxygen atoms)

### Step 4: Set Up the Ratio of Densities
Now we can set up the ratio of the densities of hydrogen to oxygen:
\[ \frac{D_{H_2}}{D_{O_2}} = \frac{P \cdot M_{H_2}}{RT} \div \frac{P \cdot M_{O_2}}{RT} \]

### Step 5: Simplify the Ratio
Since the pressure (\( P \)) and the gas constant (\( R \)) and temperature (\( T \)) are the same for both gases, they will cancel out:
\[ \frac{D_{H_2}}{D_{O_2}} = \frac{M_{H_2}}{M_{O_2}} \]

### Step 6: Substitute the Molecular Weights
Substituting the molecular weights:
\[ \frac{D_{H_2}}{D_{O_2}} = \frac{2}{32} = \frac{1}{16} \]

### Conclusion
Thus, the ratio of the densities of hydrogen to oxygen at S.T.P is:
\[ D_{H_2} : D_{O_2} = 1 : 16 \]