Density of ideal gas at 2 atm and 600K is 2g/L. Calculate relative density of this with respect to Ne(g) under similar conditions : (given : `R=1/12 atm L/mol.K`)

Options

A. 2.5
B. 2
C. 3
D. 5

Answer

To solve the problem, we will follow these steps:

### Step 1: Use the Ideal Gas Law
We start with the ideal gas equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure (in atm)
- \( V \) = volume (in L)
- \( n \) = number of moles
- \( R \) = ideal gas constant
- \( T \) = temperature (in K)

### Step 2: Express Moles in Terms of Density
The number of moles \( n \) can be expressed in terms of mass \( W \) and molar mass \( m \):
\[ n = \frac{W}{m} \]
Substituting this into the ideal gas equation gives:
\[ PV = \frac{W}{m} RT \]

### Step 3: Rearrange to Find Density
Density \( d \) is defined as mass per unit volume:
\[ d = \frac{W}{V} \]
From the ideal gas equation, we can rearrange to find:
\[ P = \frac{dRT}{m} \]
Thus, we can express the molar mass \( m \) as:
\[ m = \frac{dRT}{P} \]

### Step 4: Substitute Known Values
Given:
- Density \( d = 2 \, \text{g/L} \)
- Pressure \( P = 2 \, \text{atm} \)
- Temperature \( T = 600 \, \text{K} \)
- Gas constant \( R = \frac{1}{12} \, \text{atm L/mol K} \)

Substituting these values into the equation for molar mass:
\[ m = \frac{(2 \, \text{g/L}) \left(\frac{1}{12} \, \text{atm L/mol K}\right)(600 \, \text{K})}{2 \, \text{atm}} \]

### Step 5: Calculate Molar Mass
Calculating the above expression:
\[ m = \frac{(2)(\frac{1}{12})(600)}{2} \]
\[ m = \frac{1200}{24} \]
\[ m = 50 \, \text{g/mol} \]

### Step 6: Find Molar Mass of Neon
The molar mass of neon (Ne) is approximately:
\[ m_{Ne} = 20 \, \text{g/mol} \]

### Step 7: Calculate Relative Density
Relative density (specific gravity) is calculated as:
\[ \text{Relative Density} = \frac{m_{\text{gas}}}{m_{Ne}} \]
Substituting the values:
\[ \text{Relative Density} = \frac{50 \, \text{g/mol}}{20 \, \text{g/mol}} = 2.5 \]

### Final Answer
The relative density of the gas with respect to neon under the given conditions is:
\[ \text{Relative Density} = 2.5 \]

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