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- Show that CP = VT α (∂P/∂T) SThe 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.Estimate the values of γ = Cp,m/CV,m 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? Hint: Note that Cp,m − CV,m = R for a perfect gas.
- A¡H° = 86 kJ mol-1 ,A;S = – 140 JK-1 mol-1 , T = 299 K Express your answer as an integer.Evaluate the collision frequency between H2 molecules in a gas. Datas : 1.00 atm (101 kPa), 25 °C, σ = 0.45 nm2Calculate the work done during the isothermal reversible expansion of a gas that satisfies the virial equation of state (eqn 1C.3b) written with the first three terms. Evaluate (a) the work for 1.0 mol Ar at 273 K (for data, see Table 1C.3) and (b) the same amount of a perfect gas. Let the expansion be from 500 cm3 to 1000 cm3 in each case.
- 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?What is the contribution to the total molar energy of (a) the kinetic energy, (b) the potential energy of interaction between hydrogen chloride molecules in a gas at 298 K when 0.50 mol of molecules are confined to 1.0 dm3? Is thekinetic theory of gases justifiable in this case?Calculate the final pressure of a sample of water vapour that expands reversibly and adiabatically from 97.3 Torr and 400 cm3 to a final volume of 5.0 dm3. Take γ = 1.3.
- The heat capacity ratio of a gas determines the speed of sound in it through the formula cs = (γRT/M)1/2, where γ = Cp,m/CV,m and M is the molar mass of the gas. Deduce an expression for the speed of sound in a perfect gas of (a) diatomic, (b) linear triatomic, (c) nonlinear triatomic molecules at high temperatures (with translation and rotation active). Estimate the speed of sound in air at 25 °C. Hint: Note that Cp,m − CV,m = R for a perfect gas.Consider a gas of N fermions. (i) At T = 0, what does the Fermi momentum represent? (ii) Consider the occupation number for state i, (n;). In the limit of T → 0, what are the possible values (n;) can take? Explain your answer in terms of the relative values of the energy of the state and the chemical potential.5) Starting from the expression Cp – G, = T (P/ər), (V/ar). ' use the appropriate relations between partial derivatives to show that T(V/ar), Cp – Cy =