Calculate density of a gaseous mixture which consist of `3.01 xx 10^24` molecules of `N_2 and 32 g` of `O_2` gas at 3 atm pressure and `860 K` temperature (Given : `R=1/12` atm `L^(-1).K^(-1)`

Options

A. ` 0.6 g//L`
B. ` 1.2 g//L`
C. `0.3 g//L`
D. `12 g//L`

Answer

To calculate the density of the gaseous mixture consisting of \(3.01 \times 10^{24}\) molecules of \(N_2\) and 32 g of \(O_2\) gas at 3 atm pressure and 860 K temperature, we can follow these steps:

### Step 1: Calculate the number of moles of \(N_2\)

The number of moles can be calculated using the formula:
\[ \text{Number of moles} = \frac{\text{Number of molecules}}{\text{Avogadro's number}} \]
Given:
- Number of molecules of \(N_2 = 3.01 \times 10^{24}\)
- Avogadro's number \(= 6.022 \times 10^{23}\)

Calculating the moles of \(N_2\):
\[ \text{Number of moles of } N_2 = \frac{3.01 \times 10^{24}}{6.022 \times 10^{23}} \approx 5 \text{ moles} \]

### Step 2: Calculate the number of moles of \(O_2\)

The number of moles of \(O_2\) can be calculated using the formula:
\[ \text{Number of moles} = \frac{\text{Mass}}{\text{Molar mass}} \]
Given:
- Mass of \(O_2 = 32 \text{ g}\)
- Molar mass of \(O_2 = 32 \text{ g/mol}\)

Calculating the moles of \(O_2\):
\[ \text{Number of moles of } O_2 = \frac{32 \text{ g}}{32 \text{ g/mol}} = 1 \text{ mole} \]

### Step 3: Calculate the total number of moles

Now, we can find the total number of moles in the mixture:
\[ \text{Total number of moles} = \text{Number of moles of } N_2 + \text{Number of moles of } O_2 = 5 + 1 = 6 \text{ moles} \]

### Step 4: Use the Ideal Gas Law to find the volume

The Ideal Gas Law is given by:
\[ PV = nRT \]
Where:
- \(P = 3 \text{ atm}\)
- \(n = 6 \text{ moles}\)
- \(R = \frac{1}{12} \text{ atm L}^{-1} \text{ K}^{-1}\)
- \(T = 860 \text{ K}\)

Rearranging the equation to solve for volume \(V\):
\[ V = \frac{nRT}{P} \]
Substituting the values:
\[ V = \frac{6 \times \frac{1}{12} \times 860}{3} \]
Calculating \(V\):
\[ V = \frac{6 \times 71.67}{3} = \frac{430}{3} \approx 143.33 \text{ L} \]

### Step 5: Calculate the total mass of the gas mixture

The total mass \(m\) of the gas mixture can be calculated as:
\[ m = (\text{Number of moles of } N_2 \times \text{Molar mass of } N_2) + (\text{Number of moles of } O_2 \times \text{Molar mass of } O_2) \]
Given:
- Molar mass of \(N_2 = 28 \text{ g/mol}\)
- Molar mass of \(O_2 = 32 \text{ g/mol}\)

Calculating the total mass:
\[ m = (5 \times 28) + (1 \times 32) = 140 + 32 = 172 \text{ g} \]

### Step 6: Calculate the density of the gas mixture

Density \(\rho\) is given by:
\[ \rho = \frac{m}{V} \]
Substituting the values:
\[ \rho = \frac{172 \text{ g}}{143.33 \text{ L}} \approx 1.20 \text{ g/L} \]

### Final Answer
The density of the gaseous mixture is approximately \(1.20 \text{ g/L}\).

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