A vessel of volume `20 L` contains a mixture o hydrogen and helium at temperature of `27^(@)C` and pressure `2.0 atm` The mass of the mixture is `5 g`. Assuming the gases to be ideal, the ratio of the mass of hydrogen to heat of helium in the given mixture will be

Options

A. (a)`1 :2`
B. (b)`2 : 3`
C. (c)`2 : 1`
D. (d)`2 : 5`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert the given temperature to Kelvin
The temperature is given as \(27^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
So,
\[ T = 27 + 273.15 = 300.15 \, K \approx 300 \, K \]

### Step 2: Use the Ideal Gas Law to find the total number of moles
The Ideal Gas Law is given by:
\[ PV = nRT \]
Where:
- \(P\) = pressure in atm = 2.0 atm
- \(V\) = volume in liters = 20 L
- \(R\) = ideal gas constant = 0.0821 L·atm/(K·mol)
- \(T\) = temperature in Kelvin = 300 K

Rearranging the equation to find \(n\):
\[ n = \frac{PV}{RT} \]
Substituting the values:
\[ n = \frac{(2.0 \, \text{atm})(20 \, \text{L})}{(0.0821 \, \text{L·atm/(K·mol)})(300 \, K)} \]
Calculating \(n\):
\[ n = \frac{40}{24.63} \approx 1.62 \, \text{moles} \]

### Step 3: Set up equations for the mass of the gases
Let \(n_1\) be the number of moles of hydrogen and \(n_2\) be the number of moles of helium. The molar masses are:
- Molar mass of H₂ = 2 g/mol
- Molar mass of He = 4 g/mol

The total mass of the gas mixture is given as 5 g:
\[ n_1 \cdot 2 + n_2 \cdot 4 = 5 \quad \text{(1)} \]
We also know the total number of moles:
\[ n_1 + n_2 = 1.62 \quad \text{(2)} \]

### Step 4: Solve the system of equations
From equation (2), we can express \(n_2\) in terms of \(n_1\):
\[ n_2 = 1.62 - n_1 \]
Substituting this into equation (1):
\[ n_1 \cdot 2 + (1.62 - n_1) \cdot 4 = 5 \]
Expanding and simplifying:
\[ 2n_1 + 6.48 - 4n_1 = 5 \]
\[ -2n_1 + 6.48 = 5 \]
\[ -2n_1 = 5 - 6.48 \]
\[ -2n_1 = -1.48 \]
\[ n_1 = 0.74 \, \text{moles} \]

Now substituting \(n_1\) back into equation (2) to find \(n_2\):
\[ n_2 = 1.62 - 0.74 = 0.88 \, \text{moles} \]

### Step 5: Calculate the mass of hydrogen and helium
Now we can calculate the mass of hydrogen and helium:
- Mass of hydrogen:
\[ \text{mass of } H_2 = n_1 \cdot 2 = 0.74 \cdot 2 = 1.48 \, g \]
- Mass of helium:
\[ \text{mass of He} = n_2 \cdot 4 = 0.88 \cdot 4 = 3.52 \, g \]

### Step 6: Find the ratio of the mass of hydrogen to the mass of helium
Now we can find the ratio of the mass of hydrogen to the mass of helium:
\[ \text{Ratio} = \frac{\text{mass of } H_2}{\text{mass of He}} = \frac{1.48}{3.52} \]
Calculating the ratio:
\[ \text{Ratio} = \frac{1.48}{3.52} = \frac{2}{5} \]

### Conclusion
The ratio of the mass of hydrogen to the mass of helium in the given mixture is \( \frac{2}{5} \).