The population ratio between two energy levels separated by 1.4 ✕ 10−22 J is 0.69. What is the temperature of the system?
N2/N1 = exp(-(E2-E1) / kT)
k = boltzman constant = 1.38x10-23 J/K
0.69 = exp(-1.4x10-22 J / 1.38x10-23 J/K T)
ln 0.69 = -1.4x10-22 / 1.38x10-23 T
rearrange to get T=___K
Physical chemistry?
To determine the temperature of the system, we can use the Boltzmann equation, which relates temperature to energy levels and population ratios.
The Boltzmann equation is given by:
P1 / P2 = e^(-∆E / (k*T))
Where:
P1 is the population of the lower energy level
P2 is the population of the higher energy level
∆E is the difference in energy levels
k is the Boltzmann constant (1.38 × 10^-23 J/K)
T is the temperature in Kelvin
We are given:
P1 / P2 = 0.69
∆E = 1.4 × 10^-22 J
Plugging in these values into the Boltzmann equation, we get:
0.69 = e^(-1.4 × 10^-22 / (1.38 × 10^-23 T))
To solve for T, we need to isolate it on one side of the equation. Taking the natural logarithm (ln) of both sides can help us do that:
ln(0.69) = ln(e^(-1.4 × 10^-22 / (1.38 × 10^-23 T)))
Using the property that ln(e^x) = x, we can simplify the equation further:
ln(0.69) = -1.4 × 10^-22 / (1.38 × 10^-23 T)
Now we can solve for T. Rearranging the equation to isolate T, we have:
T = -1.4 × 10^-22 / (1.38 × 10^-23 ln(0.69))
Evaluating this expression, we find that the temperature of the system is approximately equal to:
T ≈ 309.5 K