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Use the equipartition theorem to estimate the constant-
volume molar heat capacity of (i) O3, (ii) C2H6, (iii) CO2 in the gas phase at 25 °C.
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- What are the numerical values of the heat capacities c-v and c-p of a monatomic ideal gas,in units of cal/mol.K and L.atm/mol.K?Use the equipartition theorem to estimate the constant-volume molar heat capacity of (i) I2, (ii) CH4, (iii) C6H6 in the gas phase at 25 °C.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.
- Use the equipartition principle to estimate the value of γ = Cp/CV for carbon dioxide. Do this calculation with and without the vibrational contribution to the energy. Which is closer to the experimental value at 25 °C?Use the equipartition theorem to estimate the molar internal energy of (i) I2, (ii) CH4, (iii) C6H6 in the gas phase at 25 °C.Calculate the contribution that rotational motion makes to the molar entropy of a gas of HCl molecules at 25 °C.
- Why does the isobaric molar heat capacity of most substances increase abruptly when that substance condenses from a gas to a liquid?How much energy does it take to raise the temperature of 1.0 mol H2O(g) from 100 °C to 200 °C at constant volume? Consider only translational and rotational contributions to the heat capacity.The cohesive energy density, U, is defined as U/V, where U is the mean potential energy of attraction within the sample and V its volume. Show that U = 1/2N2∫V(R)dτ where N is the number density of the molecules and V(R) is their attractive potential energy and where the integration ranges from d to infinity and over all angles. Go on to show that the cohesive energy density of a uniform distribution of molecules that interact by a van der Waals attraction of the form −C6/R6 is equal to −(2π/3)(NA2/d3M2)ρ2C6, where ρ is the mass density of the solid sample and M is the molar mass of the molecules.
- Use the equipartition principle to estimate the values of γ = Cp/CV for gaseous ammonia and methane. Do this calculation with and without the vibrational contribution to the energy. Which is closer to the experimental value at 25 °C?Calculate the vibrational, rotational, and translational contributions to the constant volume heat capacity (Cv) for 14N2 at 298 K. Assume this represents the high temperature limit for rotational energy and low temperature limit for vibrational energy. Given that Cv=20.81 J/K·mol for N2, state which type or types of energy contribute most to Cv for N2 and explain why those types of energy contribute most.The definition of enthalpy and the perfect gas equation of state can be used to estimate the standard enthalpy of ionization or electron gain and the corresponding change in internal energy. (a) Starting from Kirchhoff's law, and remembering that the molar constant-pressure heat capacity of aperfect gas is (5)/(2)R, derive an expression for the difference betweenthe change in enthalpy and change in internal energy for a gas-phase process if all species behave as if perfect gases. (b) Hence show that for ionization, ΔionHΘ - ΔionUΘ = (5)/(2)RT. (c) Use this expression to estimate the difference between the standard enthalpy of ionization of Ca(g) to Ca2+(g) and the accompanying change in internal energy at 25 °c. (d) In thesame way, show that for electron gain, ΔegHΘ - ΔegUΘ = -(5)/(2)RT.(e) Hence estimate the difference between the standard electron-gain enthalpy of Br(g) and the corresponding change in internal energy at 25 °c.