Vapour density of a gas if its density is `0.178g//L` at 1atm, 273K is

Options

A. 0.178
B. 2
C. 4
D. 0.089

Answer

To find the vapor density of a gas given its density, we can follow these steps:

### Step 1: Identify the given values
- Density of the gas (d) = 0.178 g/L
- Temperature (T) = 273 K
- Pressure (P) = 1 atm

### Step 2: Use the ideal gas equation
The ideal gas equation is given by:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = ideal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature in Kelvin

### Step 3: Relate moles to mass and molar mass
The number of moles \( n \) can be expressed as:
\[ n = \frac{m}{M} \]
where:
- \( m \) = mass of the gas
- \( M \) = molar mass of the gas

### Step 4: Substitute \( n \) in the ideal gas equation
Substituting for \( n \) in the ideal gas equation gives:
\[ PV = \frac{m}{M} RT \]

### Step 5: Rearranging the equation
Rearranging the equation to find the molar mass \( M \):
\[ PV = \frac{mRT}{M} \]
\[ PM = \frac{mRT}{V} \]

### Step 6: Express mass per volume as density
Since density \( d \) is defined as:
\[ d = \frac{m}{V} \]
we can substitute \( \frac{m}{V} \) with \( d \):
\[ PM = dRT \]

### Step 7: Solve for molar mass \( M \)
Rearranging gives:
\[ M = \frac{dRT}{P} \]

### Step 8: Plug in the values
Now, substituting the values:
- \( d = 0.178 \, \text{g/L} \)
- \( R = 0.0821 \, \text{L·atm/(K·mol)} \)
- \( T = 273 \, \text{K} \)
- \( P = 1 \, \text{atm} \)

Calculating \( M \):
\[ M = \frac{0.178 \times 0.0821 \times 273}{1} \]
\[ M = \frac{4.035}{1} \]
\[ M \approx 4.35 \, \text{g/mol} \]

### Step 9: Calculate vapor density
Vapor density is defined as:
\[ \text{Vapor Density} = \frac{M}{2} \]
So,
\[ \text{Vapor Density} = \frac{4.35}{2} \]
\[ \text{Vapor Density} \approx 2.17 \]

### Step 10: Round to the nearest whole number
Since the question asks for the vapor density and the options provided likely round to the nearest whole number, we can approximate:
\[ \text{Vapor Density} \approx 2 \]

### Final Answer
The vapor density of the gas is approximately **2**.

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