A container 11g of a `CO_(2)` and 7g of `N_(2)` at 290K . What is the density of the mixture if pressure of the mixture is 1 atmosphere . `R = 8.931 J m o l e^(-1) K^(-1)` .

Explanation

To find the density of the mixture of gases (CO₂ and N₂), we will follow these steps:

### Step 1: Calculate the total mass of the gas mixture.
The total mass \( M \) is the sum of the masses of CO₂ and N₂.

\[ M = m_{CO_2} + m_{N_2} = 11 \, \text{g} + 7 \, \text{g} = 18 \, \text{g} \]

### Step 2: Convert the total mass into kilograms.
Since density is typically expressed in kg/m³, we convert grams to kilograms.

\[ M = 18 \, \text{g} = 18 \times 10^{-3} \, \text{kg} \]

### Step 3: Calculate the number of moles of each gas.
Using the molar masses:
- Molar mass of CO₂ = 44 g/mol
- Molar mass of N₂ = 28 g/mol

Calculate the number of moles of CO₂ (\( n_{CO_2} \)):

\[ n_{CO_2} = \frac{m_{CO_2}}{M_{CO_2}} = \frac{11 \, \text{g}}{44 \, \text{g/mol}} = 0.25 \, \text{mol} \]

Calculate the number of moles of N₂ (\( n_{N_2} \)):

\[ n_{N_2} = \frac{m_{N_2}}{M_{N_2}} = \frac{7 \, \text{g}}{28 \, \text{g/mol}} = 0.25 \, \text{mol} \]

### Step 4: Calculate the total number of moles.
The total number of moles \( N \) is the sum of the moles of CO₂ and N₂.

\[ N = n_{CO_2} + n_{N_2} = 0.25 \, \text{mol} + 0.25 \, \text{mol} = 0.5 \, \text{mol} \]

### Step 5: Use the ideal gas law to find the volume of the gas mixture.
The ideal gas law is given by:

\[ PV = nRT \]

Rearranging for volume \( V \):

\[ V = \frac{nRT}{P} \]

Substituting the known values:
- \( n = 0.5 \, \text{mol} \)
- \( R = 8.931 \, \text{J/(mol K)} \)
- \( T = 290 \, \text{K} \)
- \( P = 1 \, \text{atm} = 101325 \, \text{Pa} \)

\[ V = \frac{0.5 \times 8.931 \times 290}{101325} \]

Calculating \( V \):

\[ V \approx \frac{1293.5}{101325} \approx 0.01276 \, \text{m}^3 \]

### Step 6: Calculate the density of the gas mixture.
Density \( \rho \) is defined as mass per unit volume:

\[ \rho = \frac{M}{V} \]

Substituting the values:

\[ \rho = \frac{18 \times 10^{-3} \, \text{kg}}{0.01276 \, \text{m}^3} \approx 1.41 \, \text{kg/m}^3 \]

### Final Answer:
The density of the mixture is approximately \( 1.41 \, \text{kg/m}^3 \).

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