An ideal gas with r(Gama) = 1.5 occupies a volume of 2.0 L. The gas expands isothermally to a volume of 3.0 L. Alternatively, the gas reaches the same final state by first adiabatically expanding to the final volume then heating isochorically to the final temperature. For which path is the heat transferred to the gas greatest and by what percent of the lesser amount is it greater?
To determine which path has the greater heat transfer, we need to compare the amount of heat transferred in each scenario.
Let's start with the isothermal expansion. In an isothermal process, the temperature remains constant. The heat transferred during an isothermal expansion can be calculated using the formula:
Q = nRT ln(Vf/Vi)
Where:
Q is the heat transferred,
n is the number of moles of the gas,
R is the ideal gas constant,
T is the temperature in Kelvin,
Vf is the final volume, and
Vi is the initial volume.
In this case, since we don't have the values for the number of moles or the temperature, we can simply compare the ratios of the final to initial volume for the two paths.
For the isothermal expansion, the ratio is Vf/Vi = 3.0 L / 2.0 L = 1.5.
Now, let's consider the adiabatic expansion followed by isochoric heating path. In an adiabatic process, no heat is transferred. Therefore, the heat transferred during the isochoric heating can be calculated using the formula:
Q = nCv(Tf - Ti)
Where:
Q is the heat transferred,
n is the number of moles of the gas,
Cv is the molar heat capacity at constant volume,
Tf is the final temperature, and
Ti is the initial temperature.
Since the final state is the same for both paths, the final temperature will be the same. Therefore, for the second path, we can assume the final temperature is the same as the initial temperature.
Now, let's compare the ratios of the heat transferred for the two paths.
Heat transferred in the isothermal expansion path = Q_iso = nRT ln(Vf/Vi)
Heat transferred in the adiabatic expansion + isochoric heating path = Q_ad + Q_ich = nCv(Tf - Ti)
Since both paths result in the same final state, the initial and final temperatures are the same. Therefore, Tf - Ti = 0, and the heat transferred in the second path is simply Q_ad = 0.
Comparing the two heat transfers, we have:
Q_iso/Q_ad = (nRT ln(Vf/Vi)) / 0
Division by zero is undefined, so we can conclude that the heat transferred in the isothermal expansion path is greater than the heat transferred in the adiabatic expansion + isochoric heating path. The % by which it is greater is undefined or infinite.
Therefore, the heat transferred to the gas is greatest in the isothermal expansion path, and the percentage by which it is greater is undefined or infinite.