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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): pV = Nk_B T (or pV = nRT), where N is the number of atoms, n is the number of moles, and R and k_B are ideal gas constants such that R = N_Ak_B, where N_A is Avogadro's number. In this problem, you should use Boltzmann's constant instead of the gas constant R. Remarkably, the pressure does not depend on the mass of the gas particles. Why don't heavier 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 Equipartition Theorem. The Equipartition Theorem states that the average energy associated with each degree of freedom in a system at absolute temperature T is (1/2)k_B T, where k_B = 1.38 × 10^(-23) J/K is Boltzmann's constant. A degree of freedom is a term that appears quadratically in the energy, for instance (1/2)mv_x^2 for the kinetic energy of a gas particle of mass m with velocity v_x along the x-axis. This problem will show how the ideal gas law follows from the Equipartition Theorem. To derive the ideal gas law, consider a single gas particle of mass m that is moving with speed v_x in a container with length L_x along the x direction. (Figure 1) Part A Find the magnitude of the average force (F_x) 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. Express the magnitude of the average force in terms of m, v_x, and L_x. Part B Imagine that the container from the problem introduction is now filled with N identical gas particles of mass m. The particles each have different x velocities, but their average x velocity squared, denoted (v_x^2), is consistent with the Equipartition Theorem. Find the pressure p on the right-hand wall of the container. Express the pressure in terms of the absolute temperature T, the volume of the container V (where V = L_xL_yL_z), k_B, and any other given quantities. The lengths of the sides of the container should not appear in your answer.

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):

pV = Nk_B T (or pV = nRT), where N is the number of atoms, n is the number of moles, and R and k_B are ideal gas constants such that R = N_Ak_B, where N_A is Avogadro's number. In this problem, you should use Boltzmann's constant instead of the gas constant R.

Remarkably, the pressure does not depend on the mass of the gas particles. Why don't heavier 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 Equipartition Theorem.

The Equipartition Theorem states that the average energy associated with each degree of freedom in a system at absolute temperature T is (1/2)k_B T, where k_B = 1.38 × 10^(-23) J/K is Boltzmann's constant. A degree of freedom is a term that appears quadratically in the energy, for instance (1/2)mv_x^2 for the kinetic energy of a gas particle of mass m with velocity v_x along the x-axis. This problem will show how the ideal gas law follows from the Equipartition Theorem.

To derive the ideal gas law, consider a single gas particle of mass m that is moving with speed v_x in a container with length L_x along the x direction. (Figure 1)

Part A
Find the magnitude of the average force (F_x) 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.
Express the magnitude of the average force in terms of m, v_x, and L_x.

Part B
Imagine that the container from the problem introduction is now filled with N identical gas particles of mass m. The particles each have different x velocities, but their average x velocity squared, denoted (v_x^2), is consistent with the Equipartition Theorem.
Find the pressure p on the right-hand wall of the container.
Express the pressure in terms of the absolute temperature T, the volume of the container V (where V = L_xL_yL_z), k_B, and any other given quantities. The lengths of the sides of the container should not appear in your answer.

the ideal gas law discovered experimentally is an equation of state that relates the observable state variables of the gaspressure temperature and density or quantity per volume pv nkb t or 00712

Added by Krystal S.

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