Question

An insulated rigid tank is initially evacuated. A valve is opened and atmospheric air at \(20^{\circ} \mathrm{C}, 1 \mathrm{~atm}\) enters until the pressure in the tank becomes 1 bar, at which time the valve is closed. Is the final temperature of the air in the tank equal to, greater than, or less than \(20^{\circ} \mathrm{C}\) ?

Short answer

Final temperature The exhaust air nearly remains as initial one: approx. 20°C.

Step-by-step solution

- Analyze Initial Conditions

The tank is initially evacuated, meaning it starts with no air or pressure inside. The initial pressure is therefore 0 atm and the initial temperature can be taken as irrelevant since there is no air.

- Air Enters the Tank

Air from the atmosphere at 20°C (293 K) and 1 atm enters the tank until the pressure reaches 1 bar (approx. 1 atm).

- Application of the First Law of Thermodynamics

Applying the First Law of Thermodynamics for a closed, insulated system: ... For an adiabatic and rigid tank: Q = 0) -> #0 -> -> -0 (becausae feate leak!) ->η -> -> 1->1-> nn -> ...and rigid tank (no work done): ΔUjouka ->0->=0 ->lem 2->killk == >de volume changes) > The temperature may increase or decrease if the process isn't truly adiabatic or perfectly rigid. The change in kinetic energy doesn't overcome temperature stability.

Evaluate the tank condition

Because no energy enters or leaves , tank equilibrium occurs rigidly hence temperature =atm condition

Key concepts

First Law of Thermodynamics

The First Law of Thermodynamics is often called the law of energy conservation. It states that the energy of an isolated system remains constant. This means energy cannot be created or destroyed, only transferred or converted from one form to another. For a system, it can be written as:
\( \text{dU} = \text{dQ} - \text{dW} \), where \( \text{dU} \) is the change in internal energy, \( \text{dQ} \) is the heat added to the system, and \( \text{dW} \) is the work done by the system.
Consider our problem: the tank is insulated, which means that there is no heat transfer (\( \text{dQ} = 0 \)). Also, since it's a rigid tank, no work is done by or on the system because its volume does not change (\( \text{dW} = 0 \)). Hence, the First Law of Thermodynamics reduces to \( \text{dU} = 0 \).
This means the internal energy of the air inside the tank remains constant after it enters the tank. Since internal energy is closely related to temperature for some gases, in cases like this where we consider ideal gases, the temperature will also remain constant if no energy is added or lost. In summary, the final temperature of the air in the tank does not change from its initial value of 20°C (293 K).

Adiabatic Process

An adiabatic process is one where no heat is added or removed from the system. This is exactly the case with our insulated rigid tank. Insulated means no heat transfer, making our process adiabatic.
During an adiabatic process, if the system does work, the internal energy decreases; if work is done on the system, the internal energy increases. But remember, in our rigid tank, there is no volume change, hence no work is done (\( \text{dW} = 0 \)).
Because of our adiabatic condition and no work is done, the change in internal energy \( \text{dU} \) remains zero. Therefore, in this specific scenario, the adiabatic nature of the process further confirms that the temperature inside the tank remains equal to the atmospheric temperature when the tank was filled, which is 20°C (293 K).

Rigid Tank

A rigid tank implies that the volume of the tank does not change. This has a significant impact on our calculations and understanding of the process.
In a rigid tank, since the volume is constant, there is no displacement of the boundaries of the system. This means that the work done (\( \text{dW} \)) on or by the gas inside the tank is zero because work in thermodynamics is often associated with volume changes: \( \text{Work} = P \times \text{dV}, \ \text{where} \text{dV} = 0 \)
Hence, no matter how much air is added or removed, the volume stays the same, preventing any work from being done.
Additionally, since the tank is insulated, and since we have already established that the process is adiabatic, no heat exchange occurs. Therefore, when air enters the tank and the pressure inside eventually rises to match the atmospheric pressure (1 atm), no significant temperature gradient exists to drive energy changes.
This means the temperature within the rigid tank remaining at 20°C (293 K) is consistent with both the adiabatic and the rigid nature of the system.