Final answer:
To find the work done by the gas during expansion, one would integrate p dV from V₀ to 2V₀, using the given temperature equation and ideal gas law. However, the student is likely expected to recognize the appropriate work relation from thermodynamics and select the result from the given options.
Step-by-step explanation:
The student's question asks to find the work done by an ideal gas as it expands from an initial state (2T₀, V₀) to a final state (T₀, 2V₀), with an expansion process given by T = (-T₀/V₀V + 3T₀). The work done by the gas during expansion is calculated as the area under the pressure-volume (pV) curve in a thermodynamic process.
The expression for temperature T given in the problem can be plugged into the ideal gas law, pV=nRT, to express pressure p in terms of volume V. As we have one mole of gas, n=1, and we could integrate p dV from V₀ to 2V₀ to find the work done. However, the question simplifies the calculation by providing a linear relationship between T and V.
The work done for isothermal and adiabatic expansions are different concepts and can be derived through thermodynamics principles using integration. Calculating the exact value of work done in this case requires solving the integral of p dV with the given temperature expression, applying the ideal gas law, and considering the initial and final states' temperatures and volumes. The result should be one of the provided options, and it involves entropy, heat transfer, and can be related to the Carnot cycle and isothermally or isobarically conducted processes.