3 moles of a gas are present in a vessel at a temperature of `27^@C`. What will be the value of R, the gas constant, in terms of the kinetic energy of the molecules of the gas ?

Explanation

To find the value of the gas constant \( R \) in terms of the kinetic energy of the molecules of the gas, we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and the gas constant
The kinetic energy (KE) of one mole of an ideal gas can be expressed as:
\[ KE = \frac{3}{2} RT \]
where \( R \) is the gas constant and \( T \) is the temperature in Kelvin.

### Step 2: Calculate the kinetic energy for 3 moles of gas
For \( n \) moles of gas, the kinetic energy can be expressed as:
\[ KE = \frac{3}{2} nRT \]
Given that \( n = 3 \) moles, we can substitute this into the equation:
\[ KE = \frac{3}{2} \times 3RT = \frac{9}{2} RT \]

### Step 3: Rearranging to find \( R \)
We can rearrange the equation to solve for \( R \):
\[ R = \frac{2 \times KE}{9T} \]

### Step 4: Substitute the values
We need to convert the temperature from Celsius to Kelvin. The temperature given is \( 27^\circ C \):
\[ T = 27 + 273 = 300 \, K \]
Now we can substitute \( T \) into the equation:
\[ R = \frac{2 \times KE}{9 \times 300} \]

### Step 5: Final expression for \( R \)
Thus, the expression for \( R \) in terms of kinetic energy is:
\[ R = \frac{2 \times KE}{2700} \]
This can be simplified to:
\[ R = \frac{KE}{1350} \]

### Step 6: Conclusion
The value of \( R \) in terms of the kinetic energy of the molecules of the gas is:
\[ R = \frac{KE}{1350} \]