Calculate the density of `N_(2)` gas at S.T.P. ?

Options

A. `1.250 g//L`
B. `0.628 g//L`
C. `2.450 g//L`
D. `1.42 g//L`

Explanation

To calculate the density of nitrogen gas (N₂) at standard temperature and pressure (STP), we can use the ideal gas law and derive the formula for density. Here’s a step-by-step solution:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = ideal gas constant
- \( T \) = temperature in Kelvin

### Step 2: Relate Moles to Mass
The number of moles \( n \) can be expressed in terms of mass \( m \) and molar mass \( M \):
\[ n = \frac{m}{M} \]
Substituting this into the ideal gas law gives:
\[ PV = \frac{m}{M} RT \]

### Step 3: Rearrange for Density
Density \( D \) is defined as mass per unit volume:
\[ D = \frac{m}{V} \]
From the equation \( PV = \frac{m}{M} RT \), we can express \( m \) as:
\[ m = \frac{PVM}{RT} \]
Now, substituting \( m \) into the density equation:
\[ D = \frac{PVM}{RTV} \]
This simplifies to:
\[ D = \frac{PM}{RT} \]

### Step 4: Substitute Values for STP
At STP:
- Standard pressure \( P = 1 \, \text{atm} \)
- Standard temperature \( T = 273 \, \text{K} \)
- Molar mass of nitrogen \( M = 28 \, \text{g/mol} \)
- Ideal gas constant \( R = 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \)

### Step 5: Plug in the Values
Now we can substitute these values into the density formula:
\[ D = \frac{(1 \, \text{atm})(28 \, \text{g/mol})}{(0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1})(273 \, \text{K})} \]

### Step 6: Calculate the Density
Calculating the denominator:
\[ 0.0821 \times 273 \approx 22.4143 \]
Now substituting back:
\[ D = \frac{28}{22.4143} \approx 1.250 \, \text{g/L} \]

### Final Answer
The density of nitrogen gas (N₂) at STP is approximately:
\[ D \approx 1.250 \, \text{g/L} \]