Question

For thermal equilibrium at temperature Tan appropriate measure of energy is kT where k is Boltzmann's constant. Convert the following into units of K:1 Rydberg;
103 cm- 1,

1 cm-1,

10 - 3 cm - 1;

1 eV

Answer 1

Answer:

1 cm⁻¹ =1.44K  1 ev = 1.16 10⁴ K

Explanation:

The relationship between temperature and thermal energy is

     E = K T

The relationship of the speed of light

    c =λ f = f / ν          1/λ= ν

The Planck equation is

          E = h f

Let's start the transformations

     c = f λ = f / ν        

     f = c ν

     E = h f

     E = h c ν

     E = KT

     h c ν = K T

     T = h c ν  / K =( h c / K) ν

Let's replace the constants

     h = 6.63 10⁻³⁴ J s

     c = 3 10⁸ m / s

     K = 1.38  10⁻²³ J / K

 

     v = 1 cm-1 (100 cm / 1 m) = 10² m-1

   

     T = (6.63 10⁻³⁴ 3. 10⁸ / 1.38 10⁻²³) 1 10²

     A = h c / K = 1,441 10⁻²

     T =  1.44K

     ν = 103 cm⁻¹ = 103 10² m

     T = (6.63 10⁻³⁴ 3. 10⁸ / 1.38 10⁻²³) 103 10²

     T = 148K

1 Rydberg = 1.097 10 7 m

As we saw at the beginning the λ=1 / v

     T = (h c / K) 1 /λ

     T = 1,441 10⁻²  1 / 1,097 10⁷

     T = 1.3 10⁻⁹ K

    E = 1Ev (1.6 10⁻¹⁹ J /1 eV) = 1.6 10⁻¹⁹ J

    E = KT

    T = E/K

    T = 1.6 10⁻¹⁹ /1.38 10⁻²³

    T = 1.16 10⁴ K