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- Step 1 of 7 Give the equation for the Helmholtz energy, A. (Use the following as necessary: S, T, and U.) A = U-TS U-TS Give the equation for entropy that contains the canonical partition function, Q. (Use the following as necessary: E, kB, Q, and T.) E S = kaln(Q)+ E T kB ln (Q) + 4 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 = -Tk ln(Q) + U₁ 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.) P = -(3x), ・ (a In (@)), · KB¹ KBT Step 4 of 7 For an ideal monatomic gas, the following is true. (Use the following as necessary: T, U, and V.) UT 0 Step 5 of 7 Give the equation for the canonical partition function Q. Remember that only the translational…system A with 100,000 molecules is at equilibrium at 400k with a boltzmann partition function of q=1.156518. Assume that the energy levels for system A are evenly distributed at delta U = 2Kb x T . a) calculate the probability and population distribution for the system? ( use 5 energy levels including ground state) b) calculate the entropy for the system?A 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 independent
- 2. The rotational partition function of an ethene molecule is 661 at 25°C. What is the rotational contribution to its molar entropy?N2O and CO2 have similar rotational constants (12.6 and 11.7 GHz, respect ively) but strikingly different rotational partition functions. Why?3. Calculate the residual molar entropy of a single molecule based on statistical approach if this molecule can adopt 4 orientations of equal energy at T=0K. What would be the entropy based on thermodynamic definitions?
- b. The standard molar entropy of O2 is 205.14 J mol-1 K-1 and v = 1580 cm-1 and the electronic ground state is triply degenerate. Determine the bond length of O2 from this information.Discuss the relation between the thermodynamic and statistical definitions of entropy.Part A 0.344 cm-1, Calculate the rotational partition function for SO2 at 409 K where BA = 2.03 cm¯', BB and Bc = 0.293 cm-1. (Note: the Boltzmann constant k = 1.38 × 10-23 J . K-1, speed of the light c = 3.00 x 1010 cm · s', the Planck constant h J.s). -34 6.626 x 10 Express your answer to three significant figures. ΑΣφ ? |X| X•10n qR = 4.24 • 10³ %3D Submit Previous Answers Request Answer X Incorrect; Try Again; 4 attempts remaining 圓 <8
- The translational canonical partition function Qtrs of an ideal monoatomic gas is given by: 3N 2T alnQtrs ƏT √² = Justify mathematically why the translational contribution to the molar constant-volume heat capacity is: Cy = 3Nk 3 - Nk Derive an expression for pressure in terms of the canonical partition function Q, and then obtain an expression for the Gibbs energy in terms of the canonical partition function Q. Consider the mixing of two perfectly miscible (ideal) organic solvents A and B. Calculate their respective mole fractions needed to obtain the greatest entropy of mixing.5. The vibrational modes of a particular molecule are listed below. Calculate the vibrational contribution to the molar entropy at 300 K. Mode Degeneracy wavenumber / cm-1 v1 206 v2 305 v3 1083 v4 1367 USEFUL CONSTANTS R = 8.314 J/mol/K = 0.08205 Latm/mol/K kB =1.38 x10-23 J/K h = 6.626×10-34 J s c = 3x108 m/s 1 cm-1 = 1.98x10-23 J STP = 298 K, 1bar (105 N/m²) 12 3P13C.8 Consider a system of N molecules with energy levels ɛ, = jɛ and j= 0, 1, 2,...(a) Show that if the mean energy per molecule is aɛE, then the temperature is given by B=-In 1+ Evaluate the temperature for a system in which the mean energy is ɛ, taking ɛ equivalent to 50 cm". (b) Calculate the molecular partition function q for the system when its mean energy is aɛ.