For this modified van der Waals equation (with n and b as constants), find the following two partial derivatives:
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For this modified van der Waals equation (with n and b as constants), find the following two partial derivatives:
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- Let the system be one mole of argon gas at room temperature and atmospheric pressure. Compute the total energy (kinetic only, neglecting atomic rest energies), entropy, enthalpy, Helmholtz free energy, and Gibbs free energy. Express all answers in SI units.Sketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and also for y > 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, ϕ(t) = 1 is semistable.Consider the three-dimensional harmonic oscillator, for which the potential is V ( r ) = 1/2 m ω2 r2 (a) Show that the separation of variables in Cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. Answer: En = ( n + 3/2 ) ħ ω (b) Determine the degeneracy d ( n ) of En
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- The Newton–Raphson method for finding the stationary point(s) Rsp of a potential energy surface V is based on a Taylor-series expansion around a guess, R0. Similarly, the velocity Verlet algorithm for integrating molecular dynamics trajectories [xt, vt] is based on a Taylor-series expansion around the initial conditions, [x0, v0]. Both of these methods rely strictly on local information about the system. How many derivatives do we need to compute in order to apply them?The circumference C of a circle is a function of its radius by C(r) = 2xr. Express the radius of a circle as a function of its circumference. Call this function r(C). r(C) = Preview Find r(187). r(187) = Interpret the meaning: O When the radius is 187, the circumference is r(187) O When the circumference is 187, the radius is r(187)How to evaluate the 2 partial derivatives from the expression for Z?