Question

consider a two state system at thermal equilibrium having energies 0 and 2KbT for which degree of degeneracy are 1 and 2.the value of partition function at same absolute temperature T is .............​

Answer 1

Answer:

Explanation:

Two-level systems, that is systems with essentially only two energy levels are

important kind of systems, as at low enough temperatures, only the two

lowest energy levels will be involved. Especially important are solids where

each atom has two levels with different energies depending on whether the

electron of the atom has spin up or down.

We consider a set of N distinguishable ”atoms” each with two energy levels.

The atoms in a solid are of course identical but we can distinguish them, as

they are located in fixed places in the crystal lattice. The energy of these two

levels are

ε0 and

ε1 . It is easy to write down the partition function for an atom

Z = e

−ε0 / kB T + e

−ε1 / kBT = e

−ε 0 / k BT 1+ e

−ε / kB T ( ) = Z0 ⋅ Zterm

where

ε is the energy difference between the two levels. We have written the

partition sum as a product of a zero-point factor and a “thermal” factor. This

is handy as in most physical connections we will have the logarithm of the

partition sum and we will then get a sum of two terms: one giving the zeropoint contribution, the other giving the thermal contribution.

At thermal dynamical equilibrium we then have the occupation numbers in

the two levels

n0 = N

Z

e−ε 0/kBT = N

1+ e

−ε/k BT

n1 = N

Z

e

−ε 1 /k BT = Ne −ε/k BT

1 + e−ε /k BT

We see that at very low temperatures almost all the particles are in the ground

state while at high temperatures there is essentially the same number of

particles in the two levels. The transition between these two extreme

situations occurs very roughly when

kBT ≈ε or

T ≈ θ = ε/kB , the so-called scale

temperature

θ that is an important quantity.

In this case we can directly write down the internal energy

E = n0ε 0 + n1ε 1 = N ε 0e

−ε 0/kBT + ε 1e

−ε1/kBT

e

−ε 0/kBT + e

−ε1/kBT = Nε 0 +

Nεe

−θ/T

1 + e

−θ/T

The internal energy is a monotonous

increasing function of temperature that starts

from E(0) = Nε 0 and asymptotically

approaches E(0) + Nε /2 at high

temp