Two identical samples X and Y of same gas expand from volume V to 2v exit expanded isobarically and Y is expanded isothermally if work done by sample X is w then was done by sample Y is

To determine the work done by sample Y, we need to analyze the two different processes: isobaric (constant pressure) for sample X and isothermal (constant temperature) for sample Y. Since both samples are identical and start from the same initial volume, we can derive the work done in each case using the appropriate formulas.

Understanding Work Done in Different Processes

Work done by a gas during expansion can be calculated using the formula:

• For isobaric expansion: W = PΔV
• For isothermal expansion: W = nRT ln(Vf/Vi)

Calculating Work Done by Sample X

Sample X expands isobarically from volume V to 2V. The work done (W) can be expressed as:

W_X = P(Vf - Vi)

Here, Vf = 2V and Vi = V, so:

W_X = P(2V - V) = PV

Thus, the work done by sample X is W = PV.

Calculating Work Done by Sample Y

Now, for sample Y, which expands isothermally from V to 2V, we use the isothermal work formula. Assuming the gas behaves ideally, we can express the work done as:

W_Y = nRT ln(Vf/Vi)

In this case, Vf = 2V and Vi = V, so:

W_Y = nRT ln(2V/V) = nRT ln(2)

Relating the Two Works

To relate the work done by sample Y to that of sample X, we need to consider the conditions of the gas. For an ideal gas, the relationship between pressure, volume, and temperature is given by the ideal gas law:

P = nRT/V

Substituting this into the work done by sample X, we have:

W_X = (nRT/V)V = nRT

Now we can compare the two works:

W_Y = nRT ln(2)

Since W_X = nRT, we can express W_Y in terms of W_X:

W_Y = W_X ln(2)

Final Result

Thus, if the work done by sample X is W, then the work done by sample Y can be expressed as:

W_Y = W ln(2)

This shows that the work done by sample Y is proportional to the work done by sample X, scaled by the natural logarithm of 2. This relationship highlights how different thermodynamic processes can lead to varying amounts of work done, even when starting from the same initial conditions.