To find the value of the gas constant \( R \), we can use the ideal gas equation:
\[
PV = nRT
\]
From this equation, we can rearrange it to express \( R \):
\[
R = \frac{PV}{nT}
\]
### Step 1: Identify the values needed for \( P \), \( V \), \( n \), and \( T \)
- **Pressure (\( P \))**: The standard atmospheric pressure is \( 1 \, \text{atm} \), which can be converted to Pascals:
\[
1 \, \text{atm} = 101325 \, \text{Pa}
\]
- **Volume (\( V \))**: The molar volume of an ideal gas at standard temperature and pressure (STP) is \( 22.4 \, \text{L} \). We convert this to cubic meters:
\[
22.4 \, \text{L} = 22.4 \times 10^{-3} \, \text{m}^3
\]
- **Number of moles (\( n \))**: At standard conditions, we consider \( n = 1 \, \text{mol} \).
- **Temperature (\( T \))**: The standard temperature is \( 273 \, \text{K} \).
### Step 2: Substitute the values into the equation for \( R \)
Now we substitute the values into the equation:
\[
R = \frac{(101325 \, \text{Pa}) \times (22.4 \times 10^{-3} \, \text{m}^3)}{(1 \, \text{mol}) \times (273 \, \text{K})}
\]
### Step 3: Calculate \( R \)
Calculating the numerator:
\[
101325 \, \text{Pa} \times 22.4 \times 10^{-3} \, \text{m}^3 = 2270.4 \, \text{Pa} \cdot \text{m}^3
\]
Now, divide by the denominator:
\[
R = \frac{2270.4 \, \text{Pa} \cdot \text{m}^3}{273 \, \text{mol} \cdot \text{K}} \approx 8.314 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}
\]
### Conclusion
Thus, the value of the gas constant \( R \) is:
\[
R \approx 8.314 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}
\]
