The density of a gaseous element is 5 times that of oxygen under similar conditions. If the molecule is triatomic, what will be its atomic mass ?

Options

A. 32
B. 160
C. 53.3
D. 80

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the densities of the gases.
The density of the gaseous element is given to be 5 times that of oxygen under similar conditions. Therefore, if we denote the density of oxygen as \( D_{O_2} \), the density of the gaseous element can be expressed as:
\[ D_{element} = 5 \times D_{O_2} \]

### Step 2: Use the concept of vapor density.
The vapor density (VD) of a gas is defined as half of its molar mass (molecular weight). For oxygen (O₂), the molar mass is:
\[ M_{O_2} = 16 \times 2 = 32 \text{ g/mol} \]
Thus, the vapor density of oxygen is:
\[ VD_{O_2} = \frac{M_{O_2}}{2} = \frac{32}{2} = 16 \]

### Step 3: Calculate the vapor density of the gaseous element.
Since the vapor density of the gaseous element is 5 times that of oxygen, we have:
\[ VD_{element} = 5 \times VD_{O_2} = 5 \times 16 = 80 \]

### Step 4: Relate vapor density to molar mass.
Using the relationship between vapor density and molar mass, we can express the molar mass of the gaseous element as:
\[ VD_{element} = \frac{M_{element}}{2} \]
Substituting the vapor density we found:
\[ 80 = \frac{M_{element}}{2} \]
Multiplying both sides by 2 gives:
\[ M_{element} = 80 \times 2 = 160 \text{ g/mol} \]

### Step 5: Determine the atomic mass of the triatomic molecule.
Since the gaseous element is triatomic (let's denote it as \( X_3 \)), the molar mass is the sum of the atomic masses of the three atoms:
\[ M_{element} = 3 \times \text{Atomic mass} \]
Thus, we can find the atomic mass by rearranging the equation:
\[ \text{Atomic mass} = \frac{M_{element}}{3} = \frac{160}{3} \approx 53.33 \text{ g/mol} \]

### Final Answer:
The atomic mass of the gaseous element is approximately \( 53.33 \text{ g/mol} \).

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