Thermal Physics

1.1 Gasses behaving non-ideally Real gases are non-hypothetical gases whose molecules occupy space and have interactions. The intermolecular binding (potential) energy between two molecules has been described by Lennard-Jones equation as ܸሺݎሻ ൌ 4ߝ൤ ቀఙ ௥ ቁ ଵଶ െ ቀఙ ௥ ቁ ଺ ൨ where r is the distance between particles, ε is the depth of the potential energy well, and σ is the separation distance at which the potential V is reduced to zero. (a) At what separation distance r does the minimum of the potential energy curve occur for a Lennard Jones potential? Calculate the binding energy Vmin corresponding to this point. (b) Use the Lennard-Jones potential energy to estimate the greatest net attractive force between two N2 molecules. The force acting on particles is the negative slope of the potential energy; F=−dV/dr. The Lennard Jones parameters for N2 are ε =1.268×10-21 J and σ =3.919×10-10 m. (c) It is often useful to treat a simple law that is known to be a good first approximation (in this case pVm = RT) as the first term in series in powers of a variable (in this case Vm). The so-called virial equation of state for gases is ݌ ௠ܸൌ ܴܶ ቀ1 ൅ ஻ ௏೘ ൅ ஼ ௏೘ మ ൅ ⋯ ቁ. The third virial coefficient C is less important than the second coefficient and can be ignored. The Boyle temperature TB has been defined for virial equation of state of a gas as when the gas behaves like an ideal gas (indicated by B = 0). It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out. A gas tends to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature. Recast the van der Waals equation into the virial form and then derive an equation for the Boyle temperature. Find this temperature for H2S and also the radius of the molecules of this gas. The van der Waals constants for H2S are a = 4.484 dm6 atm mol-2 and b = 0.0434 dm3 mol-1. Hint: You can assume that B/Vm << 1.0 and use the approximation (1-x) -1 = 1 + x if x <<1.0. (d) Show that the van der Waals equation leads to compressibility values of Z < 1 and Z >1, and identify the temperature range for which these values are obtained. To withstand the harsh weather of Antarctic, emperor penguins huddle in groups. Assume that a penguin is a circular cylinder with a top surface area a = 0.26 m2 and height h = 90 cm, and there are N = 1000 penguins huddling. Let ܳሶ ௣௘௡௚௨௜௡ be the rate at which an individual penguin, well separated from the rest, radiates energy to the environment (through the top and the sides). Energy loss by convection is ignored. If we can assume that the penguins huddle closely to form a huddled cylinder, what fraction of energy is saved by penguins huddling? (b) The giant hornet Vespa mandarinia japonica preys on Japanese bees, which are not defenseless like European bees. However, if one of the hornets attempts to invade a beehive, several hundred of the bees quickly form a compact ball around the hornet to stop it. They don’t sting, bite, crush, or suffocate it. Rather they overheat it by quickly raising their body temperatures from the normal 35°C to 47°- 48°C, which is lethal to the hornet but not to the bees. Assume the following: 500 bees form a ball of radius R = 1.8 cm for a time t = 18 min, the primary loss of energy by the ball is by thermal radiation, the ball’s surface has emissivity ε = 0.80, and the ball has a uniform temperature. On average, how much additional energy each bee produce during the 18 min to maintain 47°C? (8 marks) a) Write down the quantum numbers required to describe the electronic configuration of a many-electron atom. Give the possible values they can adopt. b) Specify the ground state electronic configuration of the nitrogen atom and draw its orbital level diagram. c) Which rules did you apply to determine the electronic configuration of part b)? 2.2 a) Determine the ground state electronic configuration of a Li atom. b) Using the formula for the energy levels of a hydrogen-like atom (En = Z2me4 8h2✏02n2 ), estimate the first ionisation energy of a lithium atom. Explain your answer and compare your result with the measured ionisation energy of 5.39 eV. Discuss possible physical reasons for the di↵erence between the estimated and the observed value.