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Vibrational contributions to internal energy and heat capacity
1) are temperature independent
2) are temperature dependent
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- A linear molecule may rotate about two axes. If the molecule consists of N atoms, then there are 3N- 5 vibrational modes. Use the equipartition theorem to estimate the total contribution to the molar internal energy from translation, vibration, and rotation for (a) carbon dioxide, CO2, and (b) dibromoethyne, C2Br2, at 2000 K. In contrast, a nonlinear molecule may rotate about three axes and has 3N- 6 vibrational modes. Estimate the total contribution to the molar in ternal energy from translation, vibration, and rotation for (c) nitrogen dioxide, NO2, and (d) tetrabromoethene, C2Br4,at 2000 K. In each case, first assume that all vibrations are active; then assume that none is.Which of (i) volume, (ii) heat, (iii) internal energy, (iv) density are state functions?Explain how the internal energy and entropy of a system composed of two levels vary with temperature.
- Q -A sample of argon of mass 6.56 g occupies 18.5 dm' at 305 K. (i) Calculate the work done when the gas expands isothemally against a constant external pressure of 7.7 kPa until its volume has increased by 2.5 dm. (ii) Calculate the work that would be done if the same expansion occurred reversibly.Chemistry The first excited electronic energy level of the helium atom is 3.13 ✕ 10−18 J above the ground level. Estimate the temperature at which the electronic motion will begin to make a significant contribution to the heat capacity. That is, at what temperature will 5.0% of the population be in the first excited state?Under what circumstance can the enthalpy of a system decrease for a spontaneous process?
- Answer all the questions pleaseIf HO2 and HO4H were perfect gases, and vibrational motion of the molecules was ignored, what would the value of Cp be? Plot the heat capacities of each species over the range 200 K < T < 400 K. Do you think vibrational motion is significant in these molecules? (I have the equations for Temperature dependent entropies and heat capacities for both molecules plus their enthalpies of formation values at 298.15K. I am wondering how I would begin to solve this problem.)At low temperatures the heat capacity of a metal is the sum of a contribution due to lattice vibrations (the Debye term) and a term due to the valence electrons, as given in the preceding problem. For sodium metal a = 0.507 × 10−3 J K−4 mol−1 and b = 1.38 × 10−3 J K−2 mol−1. Determine the temperature at which the Debye contribution and the electronic contribution to the entropy of sodium are equal. At higher temperatures, which contribution becomes dominant?
- The molar heat capacities for carbon dioxide at 298.0 K are shown below. C₁ = 28.95 J K-¹ mol-1 C₂ = 37.27 J K-¹ mol-1 The molar entropy of carbon dioxide gas at 298.0 K and 1.000 atm is 213.64 J K-1 mol-1. (a) Calculate the energy required to change the temperature of 1.000 mole of carbon dioxide gas from 298.0 K to 340.0 K, both at constant volume and at constant pressure. constant volume kJ kJ constant pressure (b) Calculate the molar entropy of CO₂(g) at 340.0 K and 1.000 atm. JK-1 mol-1 (c) Calculate the molar entropy of CO₂(g) at 340.0 K and 1.201 atm. JK-¹ mol-1Derivation Calculate the entropy of HBr at 298 K and 1.00 bar, given that the bond length is 1.41 Å and the masses of ¹H and 79Br are 1.008 amu and 78.92 amu, respectively. The vibrational wavenumber is 2649 cm-1. Step 1 of 7 Provide the equation needed to calculate the translational contribution to the total entropy. (Use the following as necessary: h, kB, m to represent the mass, P, π, and T.) (31) BT) = R In Strans (2μmk B¹ h³ KBT (2πmkBT) kBT P h³ P Step 2 of 7 Substitute numerical values into the equation from Step 1 to obtain a numerical calculation for the translational entropy contribution. XJ.K-1.mol-1 Strans =146.211 SubmitDerivation Calculate the entropy of HBr at 298 K and 1.00 bar, given that the bond length is 1.41 Å and the masses of ¹H and 79Br are 1.008 amu and 78.92 amu, respectively. The vibrational wavenumber is 2649 cm-¹. Step 1 of 7 Provide the equation needed to calculate the translational contribution to the total entropy. (Use the following as necessary: h, kB, m to represent the mass, P, T, and T.) (²³1) (2πmkT) h³ Strans = R In kµT (²/2) e P (2πmkBT) ³ kBT h³ P Step 2 of 7 Substitute numerical values into the equation from Step 1 to obtain a numerical calculation for the translational entropy contribution. Strans = 163.61 163.484901080763 J.K-1.mol-1 Step 3 of 7 Provide the equation needed to calculate the rotational contribution to the total entropy. (Use the following as necessary: h, I, kB, 7, σ, and T.) Srot = R In 8n²lkpT oh² + R 87² IkBT oh² Step 4 of 7 Substitute numerical values into the equation from Step 3 to obtain a numerical calculation for the rotational entropy…
