Using the ideal gas equation, determine the value of gas constant R. Given that one gram mole of a gas at S.T.P occupies a volume of 22.4 litres

Explanation

To determine the value of the gas constant \( R \) using the ideal gas equation, we will follow these steps:

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

### Step 2: Identify Given Values
We are given:
- Volume \( V = 22.4 \) liters,
- Temperature at STP \( T = 0^\circ C = 273 \) K,
- Pressure at STP \( P = 1 \) atm.

### Step 3: Convert Units
1. Convert volume from liters to cubic meters:
\[ V = 22.4 \, \text{liters} = 22.4 \times 10^{-3} \, \text{m}^3 \]
2. Convert pressure from atm to pascals:
\[ P = 1 \, \text{atm} = 1.013 \times 10^5 \, \text{N/m}^2 \]

### Step 4: Substitute Values into the Ideal Gas Equation
Since we are dealing with 1 mole of gas (\( n = 1 \)), we can simplify the ideal gas equation to:
\[ PV = RT \]
Rearranging gives us:
\[ R = \frac{PV}{T} \]

### Step 5: Substitute the Known Values
Now substitute the values of \( P \), \( V \), and \( T \) into the equation:
\[ R = \frac{(1.013 \times 10^5 \, \text{N/m}^2) \times (22.4 \times 10^{-3} \, \text{m}^3)}{273 \, \text{K}} \]

### Step 6: Calculate the Value of \( R \)
Calculating the numerator:
\[ 1.013 \times 10^5 \times 22.4 \times 10^{-3} = 2270.912 \, \text{N m} = 2270.912 \, \text{J} \]
Now divide by the temperature:
\[ R = \frac{2270.912 \, \text{J}}{273 \, \text{K}} \approx 8.31 \, \text{J K}^{-1} \text{mol}^{-1} \]

### Final Answer
The value of the gas constant \( R \) is approximately:
\[ R \approx 8.31 \, \text{J K}^{-1} \text{mol}^{-1} \]

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