Introduction:
When the volume of a gas is doubled at constant temperature, it undergoes an isothermal expansion. In this scenario, we have 3gm of nitrogen gas that doubles its volume at a constant temperature. We need to determine the change in entropy during this process.

Understanding entropy:
Entropy is a measure of the disorder or randomness in a system. In the context of gases, entropy can be thought of as the number of microstates available to the gas molecules. An increase in volume corresponds to an increase in the number of microstates, leading to an increase in entropy.

Calculating the change in entropy:
To calculate the change in entropy, we can use the equation:

ΔS = nR ln(Vf/Vi)

Where:
ΔS is the change in entropy
n is the number of moles of the gas
R is the gas constant
Vf is the final volume
Vi is the initial volume

Given values:
n = 3 gm / molar mass of nitrogen
R = Gas constant for nitrogen gas
Vi = Initial volume
Vf = 2 * Vi (since the volume doubles)

Calculating moles of nitrogen:
To calculate the number of moles of nitrogen gas, we need to know the molar mass of nitrogen, which is approximately 28 g/mol. Using this information, we can find the number of moles:

n = 3 gm / 28 g/mol

Calculating the change in entropy:
Now that we have the number of moles, we can calculate the change in entropy using the given equation:

ΔS = (3 gm / 28 g/mol) * R * ln(2 * Vi / Vi)

Simplifying the equation:

ΔS = (3 gm / 28 g/mol) * R * ln(2)

Conclusion:
In conclusion, the change in entropy when 3gm of nitrogen gas doubles its volume at constant temperature can be calculated using the equation ΔS = (3 gm / 28 g/mol) * R * ln(2). The specific value of the change in entropy will depend on the gas constant and the initial volume of the nitrogen gas.