physics-problems

• Reverse Carnot Cycle Efficiency Feb 11, 2023 - 2 min read

  An air conditioning device is working on a reverse Carnot cycle between the inside of a room at temperature \(T_2\) and the outside at temperature \(T_1\) > \(T_2\) with a monatomic ideal gas as the working medium. The air conditioner consumes the electrical power P. Heat leaks into the house according to the law \(\stackrel{.}{Q} = A(T_1 − T_2)\). Show that the efficiency of the air conditioner is \(η_{cool} = {T_2 \over T_1-T_2}\).

• Diesel Cycle Efficiency Feb 11, 2023 - 3 min read

  The Diesel cycle is the thermodynamic cycle which approximates the pressure and volume of the combustion chamber in a Diesel engine. In the Diesel cycle, the working medium in the combustion chamber is compressed adiabatically from \(V_1\), \(p_1\) to \(V_2\) < \(V_1\), \(p_2\) > \(p_1\), expanded isobarically at \(p_2\) from \(V_2\) to \(V_3\) > \(V_2\), expanded adiabatically from \(V_3\), \(p_2\) to \(V_1\) and \(p_4\) > \(p_1\), cooled isochorically at \(V_1\) from \(p_4\) to the initial pressure \(p_1\).

• The Maxwell Relations Feb 11, 2023 - 4 min read

  Many thanks to Bart Andrews for this contribution! Question Show that a relation of the kind \(f(x, y, z) = 0\) between the three quantities x, y, and z implies the relation \[\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y = -1\] between the partial derivatives. The grandcanonical partition sum \(Z_G\) is a function of the three parameters β, V, and μ. All other state variables are derived from \(Z_G\) and its partial derivatives.

• Adsorption of Molecules Onto a Surface Feb 11, 2023 - 5 min read

  Many thanks to Bart Andrews for this contribution! Question Consider a gas in contact with a solid surface. The molecules of the gas can adsorb to specific sites on the surface. These sites are sparsely enough distributed over the surface that they do not directly interact. In total, there are N adsorption sites, and each can adsorb n = 0, n = 1, or n = 2 molecules. When an adsorption site is unoccupied, the energy of the site is zero.

• Exchange of Particles Between Subsystems Feb 11, 2023 - 4 min read

  Many thanks to Bart Andrews for this contribution! Question Consider two systems \(I\) and \(II\) in contact with a common heat bath with temperature T and suppose that a mechanism exists which allows both systems to exchange particles. The probability that the composed system \(I + II\) is in a state for which system \(I\) has an energy between \(E_I\) and \(E_I+dE_I\) and a particle number \(N_I\), while \(II\) has an energy between \(E_{II}\) and \(E_{II} + dE_{II}\) with a particle number \(N_{II}\) is given by

• Diatomic Molecule as Rigid Rotor Feb 11, 2023 - 2 min read

  Download as PDF. Question Consider a molecule, such as Carbon Monoxide, which consists of two different atoms, one Carbon and one Oxygen, separated by a distance \(d\). Such a molecule can exist in quantum states of different orbital angular momentum. Each state has the energy \[ \epsilon_l={\hbar^2 \over 2I}l(l+1)\] where \(I=\mu d^2\) is the moment of inertia of the molecule about an axis through its centre of mass and \(\mu\) is the reduced mass defined by \(\frac 1\mu=\frac{1}{m_1} + \frac{1}{m_2}\).

• 2-D Polymer Bundle (Microcanonical Approach) Feb 11, 2023 - 5 min read

  This is a microcanonical ensemble approach to a simple model of a two dimensional polymer bundle. The professor of the course I took does a lot of research in the area of polymer physics and so set a few problems pertaining to them. They aren’t easily found in textbooks or online either (this website notwithstanding) but are nevertheless quite interesting in themselves. We also approached this same problem from the canonical ensemble which I’ll upload soon.

• Mixture of Two Ideal Gases Feb 11, 2023 - 4 min read

  This is a microcanonical approach to the problem of the mixture of two ideal gases. It involves a somewhat tricky integral over the surface of an N-dimensional hypersphere, and as far as I can tell is an example beloved of professors for exam questions. It might be a good one to become familiar with if you’re taking a statistical physics course. Question Enclosed in a box of volume V are \(N_1\) molecules of species 1 with mass \(m_1\) and \(N_2\) molecules of species 2 with mass \(m_2\).

• System of N Harmonic Oscillators Feb 11, 2023 - 3 min read

  The reason this general problem is so useful in a wide range of areas of physics is in physics we love to deal with harmonic approximations of systems. The point of solving the problem of N harmonic oscillators in this way is that they approximate (actually, correspond to) the behaviour of the particles in an ideal gas. Here, we work out the number of states (microstates) available to the system, the entropy of the system, and derive the energy of the corresponding ideal gas as a function of temperature.

• Random Walk of Two Drunks Feb 11, 2023 - 2 min read

  The one-dimensional random walk is a key foundational concept in statistical physics which crops up in a variety of situations, albeit normally with far more dimensions. This light-hearted question which is circling amongst statistical physics professors serves as a good introduction to random walks. If you were set this as a homework and were struggling with where to go, about now would be a good time to pat yourself on the back.

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