A polythene bag 3 litre capacity is partially filled by 1 liter of Helium gas at 0.5 atm at 300K. Subsequently, enough Ne gas is filled to make total pressure 0.6 atm at 300K. Calculate ratio of moles of Ne to He in the container.

Options

A. `1/3`
B. `1/2`
C. `1/5`
D. `1/6`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the number of moles of Helium (He)
We will use the ideal gas law equation, which is given by:

\[ PV = nRT \]

Where:
- \( P \) = pressure (in atm)
- \( V \) = volume (in liters)
- \( n \) = number of moles
- \( R \) = ideal gas constant (0.0821 atm·L/(K·mol))
- \( T \) = temperature (in Kelvin)

Given:
- Volume of the bag = 3 liters
- Volume of Helium = 1 liter (since it is partially filled)
- Pressure of Helium = 0.5 atm
- Temperature = 300 K

We can rearrange the equation to solve for \( n \):

\[ n = \frac{PV}{RT} \]

Substituting the values for Helium:

\[ n_{He} = \frac{(0.5 \, \text{atm}) \times (1 \, \text{L})}{(0.0821 \, \text{atm·L/(K·mol)}) \times (300 \, \text{K})} \]

Calculating:

\[ n_{He} = \frac{0.5}{24.63} \approx 0.0203 \, \text{moles} \]

### Step 2: Calculate the total pressure and the number of moles of Neon (Ne)
The total pressure after adding Neon is given as 0.6 atm. We can use the ideal gas law again to find the moles of Neon.

Let \( n_{Ne} \) be the number of moles of Neon. The total pressure in the container is the sum of the partial pressures of Helium and Neon:

\[ P_{total} = P_{He} + P_{Ne} \]

Where:
- \( P_{Ne} = \frac{n_{Ne}RT}{V} \)

We can express the total pressure as:

\[ 0.6 = 0.5 + \frac{n_{Ne} \times 0.0821 \times 300}{3} \]

Rearranging gives us:

\[ 0.1 = \frac{n_{Ne} \times 24.63}{3} \]

Multiplying both sides by 3:

\[ 0.3 = n_{Ne} \times 24.63 \]

Solving for \( n_{Ne} \):

\[ n_{Ne} = \frac{0.3}{24.63} \approx 0.0122 \, \text{moles} \]

### Step 3: Calculate the ratio of moles of Neon to Helium
Now we can find the ratio of moles of Neon to Helium:

\[ \text{Ratio} = \frac{n_{Ne}}{n_{He}} = \frac{0.0122}{0.0203} \approx 0.601 \]

To express this as a simplified ratio, we can multiply both the numerator and denominator by 1000 to avoid decimals:

\[ \text{Ratio} = \frac{12.2}{20.3} \]

### Final Ratio
This can be simplified further, but for practical purposes, we can express it as approximately:

\[ \text{Ratio} \approx 0.601 \]

This can also be approximated to a ratio of 1:1.66 or simplified to 1:1.5.