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1 November, 20:10

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

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Answers (1)
  1. 1 November, 23:43
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    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
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