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25)Determine the total molecular partition function for gaseous H2O at 1000 K confined to a volume of 1 cm3. The rotational constants for water are BA = 27.8 cm–1, BB = 14.5cm–1, and BC = 9.95 cm–1. The vibrational frequencies are 1615, 3694, and 3802 cm–1. The ground electronic state.
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11) List the energetic degrees of freedom expected to contribute to the internal energy at 298 K for a diatomic molecule. Given this list, what spectroscopic information do you need to numerically determine the internal energy? 12) Write down the contribution to the constant volume heat capacity from translations and rotations.
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13)In general, the high-temperature limit for the rotational partition function is appropriate for almost all molecules at temperatures above the boiling point. Hydrogen is an exception to this generality because the moment of inertia is small due to the small mass of H. Given this, other molecules with H may.
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20)Consider a collection of molecules where each molecule has two nondegenerate energy levels that are separated by 6000 cm–1. Measurement of the level populations demonstrates that there are eight times more molecules in the ground state than in the upper state. What is the temperature of the collection? .
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21) a.In this chapter, the assumption was made that the harmonic oscillator model is valid such that anharmonicity can be neglected. However, anharmonicity can be included in the expression for vibrational energies. The energy levels for an anharmonic oscillator are given by b.For H2, and Use the result from part.
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4) For which energetic degrees of freedom are the spacings between energy levels small relative to kT at room temperature? 5) For the translational and rotational degrees of freedom, evaluation of the partition function involved replacement of the summation by integration. Why could integration be performed? How does this relate back.
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18)Evaluate the vibrational partition function for NH3 at 1000 K for which the vibrational frequencies are 950, 1627.5 (doubly degenerate), 3335, and 3414 cm–1 (doubly degenerate). Are there any modes that you can disregard in this calculation? Why or why not? 19)In deriving the vibrational partition function, a mathematical expression for.
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17)Consider the following sets of populations for four equally spaced energy levels: ?/k (K) Set A Set B Set C 300 5 3 4 200 7 9 8 100 15 17 16 0 33 31 32 a.Demonstrate that the sets have the same energy. b.Determine which of the sets is the most probable. c.For the most probable set, is the distribution of energy consistent with a Boltzmann distribution? .
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1) What is the canonical ensemble? What properties are held constant in this ensemble? 2) What is the relationship between Q and q? How does this relationship differ if the particles of interest are distinguishable versus indistinguishable? 3) List the atomic and/or molecular energetic degrees of freedom discussed in this chapter. For.
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13) When are rotational degrees of freedom expected to contribute R or 3/2R (linear and nonlinear, respectively) to the molar constant volume heat capacity? When will a vibrational degree of freedom contribute R to the molar heat capacity? 14) Why do electronic degrees of freedom generally not contribute to the constant.
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7)Which species will have the largest rotational partition function: H2, HD, or D2? Which of these species will have the largest translational partition function assuming that volume and temperature are identical? When evaluating the rotational partition functions, you can assume that the high-temperature limit is valid. .
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3)At what temperature are there Avogadro’s number of translational states available for O2 confined to a volume of 1000 cm3? 4)Imagine gaseous Ar at 298 K confined to move in a two-dimensional plane of area 1.00 cm2. What is the value of the translational partition function? .
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15)Consider the following energy-level diagrams: a.At what temperature will the probability of occupying the second energy level be 0.15 for the states depicted in part (a) of the figure? b.Perform the corresponding calculation for the states depicted in part (b) of the figure. Before beginning the calculation, do you expect the temperature.
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16)Consider the following energy-level diagrams, modified from Problem 15 by the addition of another excited state with energy of 600 cm–1: a.At what temperature will the probability of occupying the second energy level be 0.15 for the states depicted in part (a) of the figure? b.Perform the corresponding calculation for the states.
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11)Imagine performing the coin-flip experiment of Problem 10, but instead of using a fair coin, a weighted coin is employed for which the probability of landing heads is twofold greater than landing tails. After tossing the coin 10 times, what is the probability of observing: a.no heads? b.two heads? c.five heads? d.eight heads? .
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15)For IF calculate the vibrational partition function and populations in the first three vibrational energy levels for T = 300 and 3000 K. Repeat this calculation for IBr Compare the probabilities for IF and IBr. Can you explain the differences between the probabilities of these molecules? .
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28)Consider an ensemble of units in which the first excited electronic state at energy ?1 is m1-fold degenerate, and the energy of the ground state is mo-fold degenerate with energy ?0. a.Demonstrate that if ?0 = 0, the expression for the electronic partition function is b.Determine the expression for the internal energy.
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22)Consider a particle free to translate in one dimension. The classical Hamiltonian is H = p2/2m. a. Determine qclassical for this system. To what quantum system should you compare it in order to determine the equivalence of the classical and quantum statistical mechanical treatments? b. Derive qclassical for a system with translational.
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14)Barometric pressure can be understood using the Boltzmann distribution. The potential energy associated with being a given height above the Earth’s surface is mgh, where m is the mass of the particle of interest, g is the acceleration due to gravity, and h is height. Using this definition of the.
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