The population ratio between two energy levels separated by 1.4 ✕ 10−22 J is 0.69. What is the temperature of the system?
To find the temperature of the system, we need to use the Boltzmann distribution equation, which relates the population ratio of two energy levels to the temperature.
The Boltzmann distribution equation is given by:
P2/P1 = e^(-ΔE / kT)
Where:
P2/P1 is the population ratio of the two energy levels,
ΔE is the energy difference between the two levels,
k is the Boltzmann constant (1.38064852 × 10^(-23) J/K),
and T is the temperature in Kelvin.
In this case, we are given P2/P1 = 0.69 and ΔE = 1.4 × 10^(-22) J.
Let's rearrange the equation to solve for T:
ln(P2/P1) = -ΔE / kT
Now, substitute the given values into the equation:
ln(0.69) = -(1.4 × 10^(-22)) / (1.38064852 × 10^(-23) × T)
Using a scientific calculator, calculate the natural logarithm of 0.69, which is approximately -0.371.
-0.371 = -(1.4 × 10^(-22)) / (1.38064852 × 10^(-23) × T)
Now, let's solve for T:
T = -(1.4 × 10^(-22)) / (-0.371 × (1.38064852 × 10^(-23)))
Simplifying the equation, we get:
T ≈ 2.50 × 10^2 K
Therefore, the temperature of the system is approximately 250 K.