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Chapter 4: Q. 4.36 (page 148)

An apparent limit on the temperature achievable by laser cooling is reached when an atom's recoil energy from absorbing or emitting a single photon is comparable to its total kinetic energy. Make a rough estimate of this limiting temperature for rubidium atoms that are cooled using laser light with a wavelength of 780 nm.

Short Answer

Expert verified

Limiting temp is 1.2 x 10-7 K

Step by step solution

01

Given Information

Wavelength , λ =780 nm.

02

Explanation

de Broglie wavelength formula is written as

p=hλ

Kinetic energy in terms of momentum , can be written as:

K=p22m

Kinetic energy in terms of temperature can be written as:

K=32kT

K=kinetic Energy
P=momentum
M= mass
k= Boltzmann constant

T=Temperature

Now substitute values and calculate

K=p22m=32kT⇒p22m=32kT⇒p23m=k

∵p=hλ⇒hλ23m=kT⇒T=h23kmλ2

⇒T=6.63×10-34Js23×85×1.66×10-27kg780×10-9m21.38064852×10-23m2kgs-2K-1

So T= 1.2 x 10-7 K

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Most popular questions from this chapter

The ingenious Stirling engine is a true heat engine that absorbs heat from an external source. The working substance can be air or any other gas. The engine consists of two cylinders with pistons, one in thermal contact with each reservoir (see Figure 4.7). The pistons are connected to a crankshaft in a complicated way that we'll ignore and let the engineers worry about. Between the two cylinders is a passageway where the gas flows past a regenerator: a temporary heat reservoir, typically made of wire mesh, whose temperature varies

gradually from the hot side to the cold side. The heat capacity of the regenerator is very large, so its temperature is affected very little by the gas flowing past. The four steps of the engine's (idealized) cycle are as follows:
i. Power stroke. While in the hot cylinder at temperature Th, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure.
ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to Tc.
iii. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in.
iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at Th.

(a) Draw a PV diagram for this idealized Stirling cycle.
(b) Forget about the regenerator for the moment. Then, during step 2, the gas will give up heat to the cold reservoir instead of to the regenerator; during step 4 , the gas will absorb heat from the hot reservoir. Calculate the efficiency of the engine in this case, assuming that the gas is ideal. Express your answer in terms of the temperature ratio Tc / Th and the compression ratio (the ratio of the maximum and minimum volumes). Show that the efficiency is less than that of a Carnot engine operating between the same temperatures. Work out a numerical example.
(c) Now put the regenerator back. Argue that, if it works perfectly, the efficiency of a Stirling engine is the same as that of a Carnot engine.
(d) Discuss, in some detail, the various advantages and disadvantages of a Stirling engine, compared to other engines.

A small scale steam engine might operate between the temperatures 20°Cand 300°C, with a maximum steam pressure of 10bars. Calculate the efficiency of a Rankine cycle with these parameters.

A common (but imprecise) way of stating the third law of thermodynamics is "You can't reach absolute zero." Discuss how the third law, as stated in Section 3.2, puts limits on how low a temperature can be attained by various refrigeration techniques.

Prove directly (by calculating the heat taken in and the heat expelled) that a Carnot engine using an ideal gas as the working substance has an efficiency of1-tcth

Prove that if you had a heat engine whose efficiency was better than the ideal value (4.5), you could hook it up to an ordinary Carnot refrigerator to make a refrigerator that requires no work input.
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