Temperature of Bose Einstein Condensation

[tanaygupta2000]

Homework Statement

An ideal gas of Rb-87 particles at 100K is compressed isothermally. Find out the number density of the gas at which Bose Einstein Condensation starts.

Relevant Equations

n = 2.612 [2πmk(Tc)/h^2]^(3/2)
n(Excited) = 2.612 [2πmkT/h^2]^(3/2)

where n = N/v
Tc = Critical temperature

For finding n, I believe that directly applying the above formula gives the results. But I am confused whether I have to put the value of given temperature T = 100K or the actual temperature Tc for Rb-87 at which BE Condensation occurs (i.e. 100 nK, which I got from a quick Google search)? Or do I need to employ a third method of calculating Tc from the available data? Please help !

 

[DrClaude]

For finding n, I believe that directly applying the above formula gives the results. But I am confused whether I have to put the value of given temperature T = 100K or the actual temperature Tc for Rb-87 at which BE Condensation occurs (i.e. 100 nK, which I got from a quick Google search)?

There is no "actual temperature" for BEC. As the equation tells you, $T_c$ is actually a function of $n = N/V$, it depends on the density of the gas. (Note also that the equation you cite is derived for a particle-in-a-box. Actual experiments are carried out in harmonic traps, which leads to a different equation for $T_c$, so even if you take the experimental value of the density $n$, tou will not get the same $T_c$.)

If your initial temperature is 100 K and you compress isothermally, what will be the temperature at the onset of BEC?

 

[tanaygupta2000]

There is no "actual temperature" for BEC. As the equation tells you, $T_c$ is actually a function of $n = N/V$, it depends on the density of the gas. (Note also that the equation you cite is derived for a particle-in-a-box. Actual experiments are carried out in harmonic traps, which leads to a different equation for $T_c$, so even if you take the experimental value of the density $n$, tou will not get the same $T_c$.)

If your initial temperature is 100 K and you compress isothermally, what will be the temperature at the onset of BEC?

Sir after Isothermal Compression, there will be no change in temperature. Therefore the final temperature at the onset of BEC will be the same, i.e., 100K.

 

[DrClaude]

Sir after Isothermal Compression, there will be no change in temperature. Therefore the final temperature at the onset of BEC will be the same, i.e., 100K.

Exactly. So you only need to calculate the corresponding density for BES at that temperature.