Answer Happy

Thermodynamic pressure p can be defined using a relationship between the energy E of a system and its volume V: JE p= av where the subscript "S" means that the entropy is kept constant while the volume is varied. Use this relationship to obtain a back-of-envelope estimate of the pressure inside a ground-state hydrogen atom. To do this, express the properties of the hydrogen atom symbolically, then treat the Bohr radius a as a hypothetical variable, so that the above pressure equation can be expressed as p= JE av da 25)/(0 da (a) Starting with the ground-state energy eigenvalue, E= ħ² 2m, a² find the derivative (OE/da). (Keep the answer in the symbolic form!) (b) Find the average radius of the hydrogen atom: 4r² dr where the ground-state wavefunction is (from the H atom lecture notes) 1 2 -rla √4π a³/¹2 Again, keep the answer in the symbolic form. Discuss why 4² is needed in the equation for (r) | (c) Find the effective volume of the atom as V = and the derivative (@V / ĉa). (d) Combine (a) and (c) to obtain the equation for the pressure. (e) Check the physical units of (d)- what are they? (f) Calculate the value of the pressure in SI units and then in atmospheres. Provide a brief discussion. (g) What is the physical meaning of the negative sign of the pressure?