Question

The ideal gas law, discovered experimentally, is an equation of state that relates the observable state variables of the gas. pressure, temperature, and density (or quantity per volume

$$ \eta V=N k_{\mathrm{B}} T(\mathrm{or} p V=n \mathrm{RT}) $$

Where \(N\) is the number of atoms, \(n\) is the number of moles, and \(R\) and \(k_{\mathrm{B}}\) are ideal gas constants such that \(R=N_{\mathrm{A}} k_{\mathrm{B}}\), where \(N_{A}\) is Avogadro's number. In this problem. you should use Boltzmann's constant instead of the gas constant \(R\).

Remaıkably. the pressure does not depend on the mass of the gas particles. Why don't heaver gas particles generate more pressure? This puzzle was explained by making a key assumption about the connection between the microscopic world and the macroscopic temperature \(T\). This assumption is called the Equibartion Theorem.

The Equipartion Theorem states that the average energy associated witn each degree of treedom in a system at absolute temperature \(T\) is \(\frac{1}{8} h_{\mathrm{B}} T_{\text {. where }} k_{\mathrm{H}} \quad 1.38 \times 10^{-2 \mathrm{~S}} \mathrm{~J} / \mathrm{K}\) is

Boltzmann's constant A degree of freedom is a term that appears quadratically in the energy, tor instance \(\frac{1}{2} m u_{x}^{2}\) tor the kinetic.

Part A

Find the magnitude of the average force \(\left(F_{z}\right\}\) in the \(x\) direction that the particle exerts on the right-hand wall of the container as it bounces back and forth. Assume that collisions between the wall and particle are elastic and that the position of the container is fixed. Be careful of the sign of your answer.

Part B

Imagine that the container from the problem introduction is now tiled with \(N\) identical gas particles of mass \(m\). The particles each have different \(x\) velocities. but their average \(x\) velocity squared, denoted \(\left(v_{x}^{2}\right)\), is consistent with the Equipartition Theorem.

Find the pressure p on the right-hand wal of the container.

Answer 1