The cohesive energy density, V, is defined as U/V, where U is the mean potential energy of attraction within the sample and V its volume. Show that \(\mathcal{U}=\frac{1}{2} \mathcal{N}^{2} \int V(R) \mathrm{d} \tau\) 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 \(-C_{6} / R^{6}\) is equal to \(-(2 \pi / 3)\left(N_{\mathrm{A}}^{2} / d^{3} M^{2}\right) \rho^{2} C_{6}\), where \(\rho\) is the mass density of the solid sample and M is the molar mass of the molecules.
Text Transcription:
U=1/2 N^2 int V(R) d tau
-C_6/R^6
-(2 pi/3)(N_A^2/d^3 M^2) rho^2 C_6
rho
17.2 • The Reaction Quotient and the Equilibrium Constant We have introduced the equilibrium constant in terms of a ratio of rate constants, but the original research on chemical equilibrium was developed many years before the principles of kinetics. In 1864, two Norwegian chemists, Cato Guldberg and Peter Waage, observed that at a given temperature, a chemical system reaches a state in which a particular ratio of product to reactant concentrations has a constant value. This is a statement of the law of chemical equilibrium, or the law of mass action. Changing Value of the Reaction Quotient The particular ratio of concentration terms that we write for a given reaction is called the reaction quotient (Q, also known as the mass-action expression). For the reversible breakdown of N O to NO , the reaction quotient, which is based directly on the 2 4 2 balanced equation, is As the reaction proceeds toward equilibrium, the concentrations of reactants and products change continually, and so does their ratio, the value of Q: at a given temperature, at the beginning of the reaction, the concentrations have initial values, and Q has an initial value; a moment later, the concentrations have slightly different values, and so does Q; after another moment, the concentrations and the value of Q change further. These changes continue, until the system reaches equilibrium. At that point, reactant and product concentrations have their equilibrium values and no longer change. Thus, the value of Q no longer changes and equals K at that temperature:
