In 12.2 we assumed that the potential energy of air...

In $\S 12.2$ we assumed that the potential energy of air molecules is infinitely large inside a bicycle pump's walls. This cannot be strictly true. Give a reasoned order-of-magnitude estimate for the potential in the walls, and consider how valid it is to approximate this by infinity.

Key Concepts

Thermal Energy Scale

The thermal energy scale, typically given by kT (where k is Boltzmann’s constant and T the absolute temperature), sets the basic energy for the motion of molecules at a given temperature. In many physical situations, if the barrier potential is much larger than kT, then the particles rarely have enough energy to overcome the barrier. This comparison is central to validating the infinite potential approximation in scenarios where the ambient thermal energy is extremely low relative to the barrier height.

Validity of the Infinite Potential Approximation

This concept involves assessing when it is acceptable to replace a very high but finite potential energy barrier with an idealized infinite potential. The approximation holds if the actual potential in the wall is many orders of magnitude larger than the energies involved in the system (such as the kinetic or thermal energy of the gas molecules). When this inequality is maintained, the probability of particle penetration into the wall is essentially zero, making the infinite potential a valid simplifying assumption.

Potential Energy Barrier

This concept refers to the energy required for a particle to penetrate or enter a region where strong repulsive forces act. In the context of solid walls, the interatomic forces create a potential energy barrier that is very high compared to the typical energies of the incident molecules. This high barrier ensures that the probability of particles crossing into the wall is negligibly small, which is why we model it with an infinite potential in simplified treatments.

Order-of-Magnitude Estimation

Order?of?magnitude estimation is a technique whereby one approximates the scale of a physical quantity by comparing it with known energy scales. Here, one might compare the potential barrier height with the thermal energy (approximately 0.025 eV at room temperature) to conclude that if the wall potential is several orders of magnitude larger (for example, tens or even hundreds of electronvolts), then the approximation to an infinite barrier is justified for practical purposes.

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