A vessel contains a mixture of `7 g` of nitrogen and `11 g` carbon dioxide at temperature ` T = 290 K`. If pressure of the mixure is `1 atm (= 1.01 xx 10^5 N//m^2)`, calculate its dencity `(R = 8.31 J//mol - K)`.

Explanation

To calculate the density of a mixture of nitrogen and carbon dioxide, we can follow these steps:

### Step 1: Calculate the number of moles of nitrogen and carbon dioxide.

1. **For nitrogen (N₂)**:
- Given mass = 7 g
- Molar mass of nitrogen (N₂) = 28 g/mol
- Number of moles (n₁) = mass / molar mass = 7 g / 28 g/mol = 0.25 mol

2. **For carbon dioxide (CO₂)**:
- Given mass = 11 g
- Molar mass of carbon dioxide (CO₂) = 44 g/mol
- Number of moles (n₂) = mass / molar mass = 11 g / 44 g/mol = 0.25 mol

### Step 2: Calculate the total number of moles in the mixture.

- Total number of moles (N_total) = n₁ + n₂ = 0.25 mol + 0.25 mol = 0.5 mol

### Step 3: Calculate the average molar mass of the mixture.

- Molar mass of the mixture (M) can be calculated using the formula:

\[ M = \frac{(n₁ \cdot M₁) + (n₂ \cdot M₂)}{N_{total}} \]

where:
- M₁ = Molar mass of nitrogen = 28 g/mol
- M₂ = Molar mass of carbon dioxide = 44 g/mol

- Plugging in the values:

\[ M = \frac{(0.25 \cdot 28) + (0.25 \cdot 44)}{0.5} \]

\[ M = \frac{(7 + 11)}{0.5} = \frac{18}{0.5} = 36 \text{ g/mol} \]

### Step 4: Use the ideal gas equation to find the density.

The ideal gas equation can be rearranged to find density (ρ):

\[ P = \rho \cdot R \cdot T / M \]

Rearranging gives:

\[ \rho = \frac{P \cdot M}{R \cdot T} \]

### Step 5: Substitute the values into the density formula.

- Given:
- Pressure (P) = 1.01 x 10^5 N/m²
- Molar mass (M) = 36 g/mol = 0.036 kg/mol (convert to kg)
- R = 8.31 J/(mol·K)
- Temperature (T) = 290 K

Substituting these values into the density equation:

\[ \rho = \frac{(1.01 \times 10^5) \cdot 0.036}{8.31 \cdot 290} \]

### Step 6: Calculate the density.

Calculating the numerator:

\[ 1.01 \times 10^5 \cdot 0.036 = 3636 \]

Calculating the denominator:

\[ 8.31 \cdot 290 = 2410.9 \]

Now, calculating density:

\[ \rho = \frac{3636}{2410.9} \approx 1.51 \text{ kg/m}^3 \]

### Final Answer:

The density of the mixture of nitrogen and carbon dioxide is approximately **1.51 kg/m³**.

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