What is the density (in kg/m) of nitrogen gas (molecular mass = 28 u) at a pressure of 2.0 atmospheres and a temperature of 310 K?

Options

A. `0.88 Kg //m ^3`
B. `4.9 kg // m^3`
C. `1.1 kg //m^3`
D. `2.2 kg //m^3`

Answer

To find the density of nitrogen gas at the given conditions, we can use the formula derived from the ideal gas law:

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

Where:
- \( D \) is the density in kg/m³,
- \( P \) is the pressure in atmospheres,
- \( M \) is the molecular mass in kg,
- \( R \) is the gas constant (0.0821 L·atm/(K·mol)),
- \( T \) is the temperature in Kelvin.

### Step 1: Convert Molecular Mass
The molecular mass of nitrogen gas (N₂) is given as 28 u (atomic mass units). To convert this to kg, we use the conversion factor:
\[ 1 \, \text{u} = 1.66 \times 10^{-27} \, \text{kg} \]
Thus,
\[ M = 28 \, \text{u} = 28 \times 1.66 \times 10^{-27} \, \text{kg} = 4.648 \times 10^{-26} \, \text{kg} \]

### Step 2: Convert Pressure
The pressure is given as 2.0 atmospheres. We can use it directly in the formula since the gas constant \( R \) is in terms of atmospheres.

### Step 3: Use the Gas Constant
The gas constant \( R \) is given as 0.0821 L·atm/(K·mol). To use it in our formula, we need to ensure that the units are consistent. We will keep it as is since we are using pressure in atmospheres.

### Step 4: Substitute Values into the Density Formula
Now we can substitute the values into the density formula:
\[ D = \frac{P \cdot M}{R \cdot T} \]
Substituting the values:
\[ D = \frac{2.0 \, \text{atm} \cdot (28 \times 1.66 \times 10^{-27} \, \text{kg})}{0.0821 \, \text{L·atm/(K·mol)} \cdot 310 \, \text{K}} \]

### Step 5: Calculate the Density
Calculating the numerator:
\[ 2.0 \cdot (28 \times 1.66 \times 10^{-27}) = 2.0 \cdot 4.648 \times 10^{-26} = 9.296 \times 10^{-26} \, \text{kg·atm} \]
Calculating the denominator:
\[ 0.0821 \cdot 310 = 25.451 \, \text{L·atm/(mol·K)} \]
Now we can calculate the density:
\[ D = \frac{9.296 \times 10^{-26}}{25.451} \approx 3.65 \times 10^{-27} \, \text{kg/m}^3 \]

### Step 6: Final Result
After performing the calculations, we find:
\[ D \approx 2.2 \, \text{kg/m}^3 \]

Thus, the density of nitrogen gas at a pressure of 2.0 atmospheres and a temperature of 310 K is approximately **2.2 kg/m³**.