Calculate the natural line-width of a transition for Cu that has a lifetime of 5 ns. Estimate the contribution of Doppler broadening to the natural line-width at 2000 K and at 8,000 K.

1)Grand free energy is defined as 0 =U-TS-µN A) Prove that in the grand canonical ensemble =-k3TlogZg. B)Using the Zg expression for the ideal gas, calculate the pressure and obtain the general state equation for the ideal gas.

Prove that (x) = 0 for the ground state of a harmonic oscillator. b) Prove that (2²) 2 uk for the ground state of a harmonic oscillator. c) Use the result in (b) to calculate the root-mean-square amplitude of 1ªN2 in its ground state. Use k = 2260 N m-1 for 14N2. The mass of 14N = 14.003 amu. d) The value you obtained in (c) is a measure of how much the bond length in this molecule varies (about its equilibrium length) due to vibrational motion. Given that the equilibrium bond length of 14N2 is 109.77 pm, by how much does the bond length vary due to vibrational motion, as a percent of its equilibrium value?

In the canonical ensemble, we control the variables T, p, and N, and the fundamental function is the Gibbs free energy (G). But if we control T, p, and μ, then we will have a different fundamental function, Z (This is the case for cells that often regulate their temperature, pressure, and chemical potentials to maintain equilibrium). Which of the below options should the Z function equal? H - TS - μN H + TS + μN H + TS - μN G + μN F - pV - μN -H + TS + μN

3) Consider the collection of identical harmonic oscillators (as in the Einstein floor). The permitted energies of each oscilator (E = nhf (n=0, 1, 2.0, hf. 2hf and so on. a) Calculate the splitting function of a single harmonic oscitor. What is the splitting function of N oscilator? wwww wwwwww www www b) Obtain the average energy of the T-temperature N oscilator from the split function. c) Calculate the heat capacity of this system and T → 0 ve T → 0 in limits, what is the heat capacity of the system? Are these results in line with the experiment? Why? What's the right theory about that? w w d) Find the Helmholtz free energy of this system. www ww e) which gives the entropy of this system as a function of temperature. ww wd wwww ww

For a classical ideal gas given the square root of p^2 in the amount a(mkt) then the value of a is

Verify the law of  addition of quantum mechanical amplitude in case of neutron diffraction in single crystal.

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The value of a partition function roughly represents the maximum energy of the states at a given temperature. O True False

......... In Maxwell Boltzman distribution, what is tne avorage kinetic energy mu)= Jmv°n(w)dv %3D S Using this result determine and