6. Show that .= V, starting with H = E + PV and an expression for dE. (Also remember H = H(S,P).)
Starting with an expression for U(S,V) , show that π(v) = (dU/dV)T is given by π(v)= (dp/dT)V - p .
Starting with an expression for U(S.V), show that m(V) = (dU/dV)T is given by Tt(v)= (dp/dT)V-P.
с V A wave packet describes a particle having momentum p. Starting with the relativistic relationship E2 = p2c2 + Eo?, show that the group velocity is ßc and the phase velocity is where ß = ). How can the phase velocity physically be greater than c? The group velocity of a wave is represented by ugr dE which gives the following expression. (Use the following as necessary: C, P, and Eo.) Ugr dE dp The phase velocity of a...
Exercise 6 subspace S of V. Show that lIpll2 (p, ) Exercise 6 subspace S of V. Show that lIpll2 (p, )
5) a) Beginning with the fundamental relation, dE = Tds - PdV, show that S = S(E, D) b) Use the answer to part a to show that: c) Legendre transform S with respect to the conjugate variables (E to obtain the Helmholtz free entropy s and show that the resulting expression for dø indicates that ° = .D Note: The expression dE = Tds - PdV is sufficient proof that E = E(S.V)
7. a) Starting with the expression for heat added to a system: dq = Cydt + (P + (**).) av and letting S = S(VT), show that the second law of thermodynamics requires that all equations of state must have the form: (P+ **)n) = ctf(V) where c is a constant and f(V) is a function only of V. b) Show that this is true for an ideal gas and a van der Waal's gas. c) For a Virial gas...
5. s a de current and In the circuit shown in Fig. P e, is a sinusoidal signal. Capacitors C, and C, are very large; their function is to couple the signal to and from the diode but block the de current from flowing into the signal source or the load (not shown). Use the diode small-signal model to show that the signal component of the output voltage is V, If v, = 10mV, find v, for 1 = 1...
If P = nRT/(V-n), then which of the following is false? A. PV = nRT + Pn B. 0 = RT + P – PV/n C. V = nPRT + nP2 D. 1 = nRT/PV + n/V E. V = (nRT/P) + n
For the ideal gas equation PV = RT, find an expression for (partial differential P/partial differential V)_T by using the method of implicit differentiation (make sure you show all your work). Compare your answer to the result you get by first solving for P in the ideal gas equation and then taking the derivative. b) Repeat part (a) for the van der Waals equation of state.
a) Discuss why the de Broglie wavelength λ corresponding to a momentum p (p wavenumber given by k # 2n/A) leads to a representation of p by the operator p as (h/) (d/dx) hk, where k is the b) Using theoperao orm of p given in part a, show that,pih c) The total energy of a simple harmonic oscillator of mass M and spring constant K can be written as H- p2/M + ke . If the mass is displaced...