6) Spectroscopy of nitrogen gas (N₂) reveals a harmonic oscillator spectrum for the vibrational modes, with €, = (r + ½)ħw, where hw = 0.3 eV. Assuming only the two lowest states are occupied, find the probability of r = 0 and r = 1 if the gas is in thermal equilibrium at 1000 K.
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- 5) Consider a substance that undergoes a phase transition at temperature T, = 300K and pressure P, = 5bar. The variation of Gibbs free energy per mole G with pressure for constant temperature T = T, in the vicinity of the phase transition is shown below. 10 600 500 6- 5- 400 300 200 3- 100 P3o 200 250 300 T(KI 400 450 350 PIbar) (Left) Variation of Gibbs free energy with pressure in the vicinity of a first-order phase transition at T, = 300K (solid line). The dashed line indicates the location of the phase transition. (Right) Phase boundary in the T-P plane (solid line). The dashed lines indicate the location of the reference point T,, Po- [Ibar = 10 Pa] From the figures above, estimate the difference in volume AV between the two phases a, (per mole of substance). (Hint: dG = V dP – SdT may be useful) From the result of (a) and from Figure 2, estimate the entropy difference AS between the two phases, and from this compute the latent heat of transition L (both per mole of substance).…a) Show that for enthalpy and Gibbs free energy, H = G + TS can be written. b) Based on the exact differential of Gibbs free energyA gas of identical diatomic molecules absorbs electromagnetic radiation over a wide range of frequencies. Molecule 1, initially in the υ = 0 vibrational state, makes a transition to the υ = 1 state. Molecule 2, initially in the υ = 2 state, makes a transition to the υ = 3 state. What is the ratio of the frequency of the photon that excited molecule 2 to that of the photon that excited molecule 1? (a) 1 (b) 2 (c) 3 (d) 4 (e) impossible to determine
- 1- An ideal gas consisting of N mass points is contained in a box of volume V. Find the number of states (phase integral) A,(E) classically and, using it, derive the equation of state. (Hint: The volume of a unit sphere is a n-dimensional space is equal to1.3 Statistical Mechanics26) A system of two moles of an ideal gas increases its internal energy by 1000 J as it increases in temperature from 300 K to 340 K. There are no changes in bond energy. Which of the following statements is true? A: This gas is monatomic. B: Not enough information given; you need to know how much work was done on the gas. C: This gas is diatomic, with its vibrational modes unfrozen. D: Not enough information given; you need to know how much heat was transferred to the gas. E: This gas is diatomic, with its vibrational modes frozen out.
- Q2/ For the harmonic oscillator in classical limit, if the partition function is given by Z(B) = 1/(ßħw)N, find the entropy and internal energy. Q3/ Prove that F = -KTlnz .- Calculate the average number of phonons occupying a vibrational mode with angular fre- quency w = 4.0 x 10¹2 s-1 at T = 300 K. - Calculate the total energy of the mode at this temperature, expressing your answer in meV.Ggg
- The exact differential for the Gibbs energy is given by dG = -SdT + VdP. The form of this differential implies which of the following relationships? O (7),- (), ƏG Әт ƏG др P T (37), - - (-), P T as ( x) - (*), = др T (327), = -(0)₁ == P TA diatomic gas molecule can be in one of two vibrational energy levels, separated by 0.1 eV. Give the probabilities to be in either state and use these to calculate their relative populations at room temperature, T≈ 300 K. [You may use that kB ≈ 8.6 × 10−5 K eV−1]4) A surface with No adsorption centers has N (S N,) gas molecules adsorbed on it. Show that the chemical potential of the adsorbed molecules is given by u = kTIn- N (No-N)a(T) where a(T) is the partition function of a single adsorbed molecule. Solve the problem by constructing the grand partition function as well as the partition function of the system. [Neglect the intermolecular interaction among the adsorbed molecules.]