3. Consider an ideal gas of N particles, each of mass M, confined to a one- dimensional line of l...
3. Consider a hypothetical non-ideal gas of particles confined to exist along a line in one dimension. The particles are in thermal equilibrium but due to their complex interactions the velocity distribution function is not Maxwellian, but rather has the form: where C and vo are constants. Note that v is the velocity (not the speed) and can take on negative values. Express your answers below in terms of vo- a. Solve for the constant C b. Draw a sketch...
1. N identical particles of mass m is confined to move only about a surface of area A, and thus can be considered as a “two-dimensional gas.” In the classical limit what is the single particle partition function of one of the particles in this ideal two-dimensional gas? What is the partition function of the N particle gas? What is the mean thermal energy of this gas, and what is the entropy of this gas?
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i has momentum pi. Assume that the particles are non-interacting (ideal gas) and distinguishable. a) (2P) Calculate the canonical partition function N P for the N-particle system. Make sure to work out the integral. b) (2P) Calculate the free energy F--kBTlnZ from the partition function Z. Is F an extensive quantity? c) (2P) Calculate the entropy S F/oT from...
Pb2. Consider the case of a canonical ensemble of N gas particles confined to a t rectangular parallelepiped of lengths: a, b, and c. The energy, which is the translational kinetic energy, is given by: o a where h is the Planck's constant, m the mass of the particle, and nx, ny ,nz are integer numbers running from 1 to +oo, (a) Calculate the canonical partition function, qi, for one particle by considering an integral approach for the calculation of...
2. Consider we have put an ideal Fermi gas with (N) average particles of spin and mass m in 2D space of area A at finite temperature T. Derive the fermi energy ef as a function in temperature. Hint: vou will need to know that Ep =
2. Consider we have put an ideal Fermi gas with (N) average particles of spin and mass m in 2D space of area A at finite temperature T. Derive the fermi energy ef...
Thermodynamics
Consider an insulated container of volume V2. N ideal gas molecules are initially confined within a sub-volume (V1) by a piston and the remaining volume V2 - Viis in vacuum. Let T., P., U1, S1, A1, H1, and G1 be the temperature, pressure, internal energy, entropy, Helmholtz free energy, enthalpy, and Gibbs free energy of the ideal gas at this state, respectively. Now, imagine that the piston is removed so that the gas has volume V2. After some time...
1) Consider a particle with mass m confined to a one-dimensional infinite square well of length L. a) Using the time-independent Schrödinger equation, write down the wavefunction for the particle inside the well. b) Using the values of the wavefunction at the boundaries of the well, find the allowed values of the wavevector k. c) What are the allowed energy states En for the particle in this well? d) Normalize the wavefunction
P7D.6 Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction y,. (a) Without evaluating any integrals, explain why(- L/2. (b) Without evaluating any integrals, explain why (p)-0. (c) Derive an expression for ) (the necessary integrals will be found in the Resource section). (d) For a particle in a box the energy is given by En =n2h2 /8rnf and, because the potential energy is zero, all of this...
A one-dimensional particle of mass m is confined within the region 0 < x < a and wave function V(x, t) = sin(TI)e-iwt. a Given the wave function 1(x, t) above, show that V is independent of t. b Calculate the probability of finding the particle in the interval a 5 x 54