The population ratio between two energy levels separated by 1.4 ✕ 10−22 J is 0.69. What is the temperature of the system?

0.037 27 K

To find the temperature of the system, we can use the Boltzmann distribution equation. The Boltzmann distribution gives the population ratio of two energy levels at a given temperature.

The Boltzmann distribution equation is given by:

P2/P1 = e^(-ΔE/kT)

Where:
P1 and P2 are the population ratios of the two energy levels,
ΔE is the energy difference between the two levels (1.4 × 10^(-22) J in this case),
k is the Boltzmann constant (1.38 × 10^(-23) J/K),
T is the temperature of the system (what we need to determine).

In this case, we know the population ratio is 0.69 and the energy difference is 1.4 × 10^(-22) J. By rearranging the equation and solving for T, we can find the temperature.

Let's substitute the known values into the equation and solve for T:

0.69 = e^(-(1.4 × 10^(-22))/(1.38 × 10^(-23) × T))

Now, to solve for T, we can take the natural logarithm of both sides to remove the exponential term:

ln(0.69) = -(1.4 × 10^(-22))/(1.38 × 10^(-23) × T)

Solving for T:

T = -(1.4 × 10^(-22))/(1.38 × 10^(-23) × ln(0.69))

Calculating this value, we find T ≈ 7,263.28 K.

Therefore, the temperature of the system is approximately 7,263.28 Kelvin.