Do Path-Dependent Processes Affect Thermodynamic Quantities in Ideal Gases?

[painfive]

An ideal gas changes state from P=32, V=1 to P=1 V=8 via three different paths: first pressure then volume, first volume then pressure, and adiabatically. I need to calculate the change in heat energy, work done by the system, and change in internal energy for all three paths. Will these be the same for all the paths? What equations should I use? So far I found the heat change for the adiabatic system (zero) and the work done by the other two (only along the parts where the volume changes). I'm pretty sure the internal energy change is the same for all of them, so I need at least one more value before I can get the rest.

 

[Astronuc]

first pressure then volume, first (second?) volume then pressure, and adiabatically.

Does first pressure then volume mean first pressure changes (decreases) and then volume changes (increases)?

How about showing some equations.

e.g. $W_{1 \rightarrow 2} = \int_{V_1}^{V_2} p\, dV$

and for an adiabatic system $pV^\gamma = const$

and what is the relationship between energy and work in the gas?

 

[quicknote]

I would like to clarify some points before providing a response. Firstly, the term "ideal gas" refers to a theoretical gas that follows the ideal gas law, which assumes no intermolecular forces and perfect gas behavior. Therefore, the state of an ideal gas is fully described by its pressure, volume, and temperature. Secondly, the change in internal energy for an ideal gas is only dependent on its temperature and can be calculated using the first law of thermodynamics, which states that the change in internal energy is equal to the heat added to the gas minus the work done by the gas.

Now, to answer the question, the change in heat energy, work done by the system, and change in internal energy will not be the same for all three paths. This is because the heat and work depend on the path taken by the gas, while the change in internal energy only depends on the initial and final states.

To calculate the change in heat energy, you can use the equation Q = nCΔT, where Q is the heat added, n is the number of moles of gas, C is the molar heat capacity at constant pressure, and ΔT is the change in temperature. For the first two paths, you can calculate the change in temperature using the ideal gas law (P1V1/T1 = P2V2/T2). For the adiabatic path, since no heat is exchanged, the change in temperature will be zero.

The work done by the gas can be calculated using the equation W = -∫PdV, where P is the pressure and V is the volume. For the first path (pressure then volume), the work done will be equal to the area under the curve on a P-V diagram. For the second path (volume then pressure), the work done will be equal to the area above the curve on a P-V diagram. And for the adiabatic path, the work done will be zero.

Finally, as mentioned earlier, the change in internal energy will be the same for all three paths and can be calculated using the equation ΔU = nCΔT. You can use the change in temperature calculated from the ideal gas law to find the change in internal energy.

In summary, to calculate the change in heat energy, work done by the system, and change in internal energy for an ideal gas undergoing a change in state, you will need to use the ideal gas law