Consider a system of two energy levels separated by an energy equal to ε1. The lowest energy level is nondegenerate, and the second energy level has a degeneracy of 4. a) Write the expression for the partition function. b) Write the expression for E. Determine the value of E in the infinite temperature limit. c) Write the expression for S. Determine the value of S in the infinite temperature limit.
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- 1. Show that the partition function for a harmonic oscillator in the x direction only is given by: qx exp(-ẞhv) 1-exp(-ẞhv)E e q kT where k is The probability of a molecule having energy E is P (E) Boltzmann's constant (1.38 * 10¯²³J/K) and q is the partition function. A system has three possible energy levels: 0 cm³¹, 100 cm³¹, and 250 cm³¹. The temperature is 300 K. a) Convert each energy level to J/molecule. (Double check your units!) b) Find q, the partition function (sum up the unnormalized probabilities). The degeneracy of all levels = 1 c) Calculate the average energy (expectation value) of a molecule in units of cm³¹. Make sure to use normalized probabilities. d) Plot P vs E. Make sure to use normalized probabilities.Give the equation for the Helmholtz energy, A. (Use the following as necessary: S, T, and U.) A = U-TS - TS Give the equation for entropy that contains the canonical partition function, Q. (Use the following as necessary: E, kB, Q, and T.) E E S = k ln(Q) + B kB ln (Q) + Step 2 of 7 We only need to consider the translational translational partition function for an ideal monatomic gas, so E = U - Uo. Combine this equation with the equations for S and A from Step 1. (Use the following as necessary: KB, Q, T, U, and Up.) A = |— Tkôln(Q) + U U₁ - KBT ln(Q) Step 3 of 7 Substitute the equation from Step 2 into the given equation for P and complete the partial derivative. (Use the following as necessary: kB, Q, T, U, and Up.) -(SA), P = - a In(Q)). KBT = KBT Step 4 of 7 For an ideal monatomic gas, the following is true. (Use the following as necessary: T, U, and V.) (U) = UT - 0 Step 5 of 7 Give the equation for the canonical partition function Q. Remember that only the translational…
- The answers may be one or more than one As discussed in one of the videos, the equilibrium state of a gas in a box with a movable partition is determined by maximizing Ωtotal. Choose the expressions that correctly describe Ωtotal in this situation. a. A function of the volume V1 in the left half of the box. b. A function of the total volume V of the box. c. The total number of microstates in the left half of the box. d. The total number of microstates in the system and surroundings. e. The total number of microstates available to the system. f. The total number of microstates in the right half of the box.Calculating Partition Functions. Consider the acetylene molecule, which is linear and has the chemical structure H−C≡C−H. Calculate or determine the following: a. Symmetry number (?) and degrees of freedom b. Moment of inertia (I) and rotational temperature (Θr) c. Vibrational temperatures (Θv,i) d. Translational (qt), rotational (qr), vibrational (qv), and electronic (qe) partition functions at 298 K. e. Total partition function, (q/V). Helpful information: mass of H atom = 1.007947 amu mass of C atom = 12.01078 amu C≡C bond length = 1.203 Å C−H bond length = 1.060 Å vibrational frequencies = 1975, 3370, 3277, 729 (2), 600 (2) cm-1 (numbers in parenthesis indicate the degeneracy of that mode) ground state electronic degeneracy = 1The table includes macrostates characterized by 4 energy levels (&) that are equally spaced but with different degrees of occupation. a) Calculate the energy of all the macrostates (in joules). See if they all have the same energy and number of particles. b) Calculate the macrostate that is most likely to exist. For this macrostate, show that the population of the levels is consistent with the Boltzmann distribution. macrostate 1 macrostate 2 macrostate 3 ε/k (K) Populations Populations Populations 300 5 3 4 200 7 9 8 100 15 17 16 0 33 31 32 DATO: k = 1,38×10-23 J K-1
- Please don´t use partition equationA certain molecule can exist in either a nondegenerate singlet state or a triplet state (with degeneracy of 3). The energy of the triplet exceeds that of the singlet by ε. When ?=??(where T is a set value, i.e., ? is a constant), calculate the values of the molecular partition function, molar heat capacity, and molar entropy. Assume the molecules are distinguishable and independentWhich if the following statement describe properties of the Boltzmann expression in regards to the relative occupancy of energy states: Select one: O a. Lower temperatures favours more molecules in the higher energy levels O b. Lower the energy separation, equates to more molecules in the lower energy levels c. Lower the energy separation, equates to more molecules in the higher energy levels O d. Higher temperatures favours more molecules in the lower energy levels
- The internal energy of a system A None of these В is the sum of the rotational, vibrational, and translational energies of all of its components refers only to the energies of the nuclei of the atoms of the component molecules D is the sum of the kinetic energy of all of its components E) is the sum of the potential and kinetic energies of the componentsWe discussed in class (several times) how the Boltzmann distribution can be used to relate the relative populations of two states differing in energy by AU. Suppose you are given a vial containing a solution of glucose in water (don't ask why this would happen). For the purpose of this question, glucose exists in one of two conformations-"chair" or "boat"-with an energy difference (AU) of 25.11 kJ mol1 between them. 1. What would be the proportion of molecules in the "boat" conformation at 310K? 2. Thinking back to our discussion of the individual sources of energy that go into the potential energy calculation for a molecule (e.g. Upond Uangle, Uelectrostatic. etc), give a plausible explanation of why the "boat" conformation is less stable. H он "Chair" OH "Вoat" но но но- HO. H. HO. HO H. HO. OHQ2. Statistical Thermodynamics. Answer all parts. i. What do you understand by the term molecular partition function? i. Write the partition function for a two-level system, the lower state (at energy 0) being non-degenerate, and the upper state (and an energy ɛ) doubly degenerate. A certain molecule has a doubly degenerate excited state lying at 360 cm- above the non-degenerate ground state. At what temperature will 15% of the molecules be in the upper state? iii. iv. Calculate the translational partition function at 300 K of a molecule of molecular mass 120 g mol- in a container of volume 2.0 cm. 3 3 2 7k 9 trs (MT)?V hNA V. Given that the vibration heat capacity has the formula: u'e" Cyvib,m = R (e" – 1)? hcv 0vib where: u = kT %3D T calculate the vibrational heat capacity of chlorine (Cl2) at 1000 K given that its characteristic vibrational temperature O, is 800 K. vi. Given that the molar entropy has the form: и S. vib,m -e (e" –1) Calculate the vibrational entropy of iodine (Cl2)…