Answer:
a) P_α = exp (-ΔE / kT), b) P_β = 0.145
, d) ΔE = 309.7 meV
Explanation:
The expression for the number of molecules or particles in a given state in Boltzmann's expression
n = n₀ exp (-ΔE / kT)
Where k is the Bolztmann constant and T the absolute temperature
The probability is defined as the number of molecules in a given state over the total number of particles
P = n / n₀ = exp (- ΔE / kT)
Let's apply this expression to our case
a) P_α = n_α / n₀ = exp (-ΔE / kT)
b) the Boltzmann constant
k = 1,381 10⁻²³ J / K (1 eV / 1.6 10⁻¹⁹ J) = 8.63 10⁻⁵ eV / K
kT = 8.63 105 300 = 2,589 10⁻² eV
P_β = exp (- 50 10⁻³ /2.589 10⁻² = exp (-1.931)
P_β = 0.145
c) If the temperature approaches absolute zero, the so-called is very high, so there is no energy to reach the excited state, therefore or all the molecules go to the alpha state
d) For molecules to spend ¼ of the time in this beta there must be ¼ of molecules in this state since the decay is constant.
P_β = ¼ = 0.25
P_β = exp (- ΔE / kT)
ΔE = -kT ln P_β
ΔE = - 2,589 10⁻² ln 0.25
ΔE = 0.3097 eV
ΔE = 309.7 meV
