FrontierScience / cfa05761-b488-4491-9f77-8854ece1cf98 / A rectangular tank of width `w` along the `x` - axis is filled with a solution with concentration `C(z,t)`, where z is the vertical direction. The solution is in equilibrium. The c…

A rectangular tank of width `w` along the `x` - axis is filled with a solution with concentration `C(z,t)`, where z is the vertical direction. The solution is in equilibrium. The concentration at the bottom of the tank, `z=0,` is `C₀` and the tank is tall enough so the concentration at the top is C_top≪C₀ Let the left hand edge of the tank be at x=0 and the right hand edge at x=w.

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Outside the tank, at x=−d, we place a laser, which shines light through the tank from left to right. The laser is angled up at an angle θ above the x axis. After exiting the tank at x=w, the laser continues on to a screen at x=w+D.

The relation between the refractive index of the solution and its concentration is `n=1+ξC`, where `ξ` is a constant. 

Assume that w and θ are both small enough that we can think of the refractive index as constant along the laser's path through the solution. Ignore any refraction caused by the walls of the tank. 

Find the height z_(i) of the point of laser light on the screen. Assume that the laser light begins at height z=0 when it leaves the laser.

Give your answer as a function of `θ,d,D,ξ,k_(B),T,η,a,ρ₀,ρ,` and g, where k_(B) is Boltzmann's constant and T is the temperature of the solution, which we assume to be uniform throughout the solution. `η` is the viscosity of the solution.  a is the radius of the particles, which we assume to be spheres of the same size and mass. ρ₀ is the density of the solvent. ρ is the density of the solute. g is local gravitational acceleration, assumed to be uniform throughout the system.

`Assume that the laser light refracts upon entering and exiting the tank, but no other effects cause the laser light to deviate from a straight path.`

`Give your answer to first order in θ, i.e., in your final answer, replace any factors of sinθ or tanθ with θ and drop any terms proportional to θ²,sin²θ, or tan²θ.`

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

A rectangular tank of width `\( w\)` along the `\( x\)` - axis is filled with a solution with concentration `\(C(z,t)\)`, where \(z\) is the vertical direction. The solution is in equilibrium. The concentration at the bottom of the tank, `\( z=0\),` is `\( C_{0}\)` and the tank is tall enough so the concentration at the top is \(C\_{top} \ll C_0\) Let the left hand edge of the tank be at \(x = 0\) and the right hand edge at \(x = w\).

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Outside the tank, at \(x = -d,\) we place a laser, which shines light through the tank from left to right. The laser is angled up at an angle \(\theta\) above the \(x\) axis. After exiting the tank at \(x = w,\) the laser continues on to a screen at \(x = w + D.\)

The relation between the refractive index of the solution and its concentration is `\( n=1+\xi C\)`, where `\( \xi \)` is a constant. 

Assume that \(w\) and \(\theta\) are both small enough that we can think of the refractive index as constant along the laser's path through the solution. Ignore any refraction caused by the walls of the tank. 

Find the height \(z_i\) of the point of laser light on the screen. Assume that the laser light begins at height \(z = 0\) when it leaves the laser.

Give your answer as a function of `\( \theta, d, D, \xi, k_B, T, \eta, a, \rho_0, \rho,\)` and \(g\), where \(k_B\) is Boltzmann's constant and \(T\) is the temperature of the solution, which we assume to be uniform throughout the solution. `\( \eta \)` is the viscosity of the solution.  \(a\) is the radius of the particles, which we assume to be spheres of the same size and mass. \(\rho_0\) is the density of the solvent. \(\rho\) is the density of the solute. \(g\) is local gravitational acceleration, assumed to be uniform throughout the system.

`Assume that the laser light refracts upon entering and exiting the tank, but no other effects cause the laser light to deviate from a straight path.`

`Give your answer to first order in \(\theta,\) i.e., in your final answer, replace any factors of \(\sin\theta\) or \(\tan\theta\) with \(\theta\) and drop any terms proportional to \(\theta^2, \sin^2\theta,\) or \(\tan^2\theta\).`

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.