The study of thermodynamics is filled with subtle traps and beautiful logical deductions. This problem is a classic example of a "free expansion" scenario, which tests your fundamental understanding of the First Law of Thermodynamics and the properties of an ideal gas. Let's break down the physical reality of what is happening inside this vessel.
The Setup
A Box in Isolation
Imagine a gas confined in a chamber. The problem explicitly states that the vessel is thermally insulated. This is our first major clue.
Thermal insulation means that the walls of the container are adiabatic boundaries. No heat can flow into the system from the surroundings, and no heat can escape from the system.
Mathematically, this translates to a very simple condition for the heat exchange:
This confirms that the process is adiabatic. Option (A) is absolutely correct.
The Vacuum
Expanding into Nothingness
Now, let's look at the right side of the piston in the initial state. The external pressure is given as Pext=0. This means the gas is expanding into a perfect vacuum.
When a gas expands, it usually has to push against the atmosphere or a weight, which requires it to do work. However, in this case, there is nothing pushing back!
This phenomenon is known as free expansion. Because there is no opposing force, the gas expends no energy to expand. Therefore, the work done by the gas is zero:
The First Law
The Ultimate Balancer
With both heat and work determined, we can invoke the First Law of Thermodynamics. This law is essentially the principle of conservation of energy for thermodynamic systems.
The First Law states that the change in internal energy of a system is equal to the sum of the heat added to it and the work done on it:
Since we have already established that q=0 and W=0, the substitution is trivial but profound:
The internal energy of the gas remains perfectly constant during this irreversible expansion.
The Ideal Gas Secret
Here is where the specific nature of the gas becomes crucial. The problem specifies that we are dealing with an ideal gas.
For an ideal gas, there are no intermolecular forces of attraction or repulsion. Because of this, the internal energy does not depend on the volume or pressure of the gas. It is a strict, exclusive function of its absolute temperature.
The relationship is given by:
Since we just proved that ΔU=0, and neither the number of moles n nor the molar heat capacity Cv can be zero, it must be true that the change in temperature is zero:
This means the initial and final temperatures are identical. Thus, T1=T2, making option (B) correct. The process, despite being adiabatic, is also isothermal!
The Final State
We have established that the temperature remains constant throughout the expansion. For an ideal gas at a constant temperature, Boyle's Law governs the relationship between pressure and volume.
Boyle's Law states that the product of pressure and volume is a constant:
This confirms that option (C) is also correct. The pressure drops exactly in proportion to the increase in volume.
The Trap
Reversible vs Irreversible
Finally, we must address option (D), which suggests P2V2γ=P1V1γ. This is a very common pitfall for students.
While it is true that the process is adiabatic (q=0), the equation PVγ=constant is derived under the strict assumption that the process is reversible.
Free expansion is a highly irreversible process. You cannot spontaneously compress the gas back into its original volume without doing external work. Because the process is irreversible, the reversible adiabatic equation simply does not apply. Therefore, option (D) is incorrect.
This problem beautifully demonstrates how free expansion of an ideal gas is simultaneously adiabatic and isothermal, yet follows neither of their reversible path equations!
