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 determine the temperature of the system, we can use the Boltzmann distribution formula, which states that the ratio of the population of two energy states is given by:
P2 / P1 = e^(-∆E / kT)
Where P1 and P2 are the population ratios of the two energy states, ∆E is the energy difference between the two states, k is the Boltzmann constant (1.38 × 10^-23 J/K), and T is the temperature in Kelvin.
In this case, we are given the population ratio (P2 / P1 = 0.69) and the energy difference (∆E = 1.4 × 10^-22 J). We can rearrange the formula to solve for T:
T = -∆E / (k * ln(P2 / P1))
Let's calculate the temperature now:
T = - (1.4 × 10^-22 J) / (1.38 × 10^-23 J/K * ln(0.69))
First, we calculate ln(0.69):
ln(0.69) ≈ -0.367724785
Now we substitute the values into the equation:
T ≈ - (1.4 × 10^-22 J) / (1.38 × 10^-23 J/K * -0.367724785)
T ≈ 286.1 K
Therefore, the temperature of the system is approximately 286.1 Kelvin.