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 can use the Boltzmann distribution formula, which relates the population ratio to the energy difference and temperature.
The Boltzmann distribution formula is given as:
R = exp(-ΔE / (k*T))
Where:
- R is the population ratio between two energy levels,
- ΔE is the energy difference between the two levels,
- k is the Boltzmann constant (1.38 x 10^-23 J/K),
- T is the temperature in Kelvin.
In this case, the population ratio R = 0.69, and the energy difference ΔE = 1.4 x 10^-22 J. We need to solve for T.
Rearranging the formula, we have:
T = -ΔE / (k * ln(R))
Let's substitute the given values and calculate the temperature:
To find the temperature of the system, we can use the Boltzmann distribution equation, which relates the energy levels to temperature and the population ratio.
The Boltzmann distribution equation is given by:
P2/P1 = e^(-ΔE/kT)
Where:
P2/P1 is the population ratio between energy levels, which is given as 0.69.
ΔE is the energy difference between the two levels, which is given as 1.4 × 10^-22 J.
k is the Boltzmann constant, which is approximately 1.38 × 10^-23 J/K.
T is the temperature of the system (what we need to find).
Now we can substitute the given values into the equation:
0.69 = e^(-1.4 × 10^-22 J / (1.38 × 10^-23 J/K * T))
To solve for T, we need to isolate it. Taking the natural logarithm (ln) of both sides of the equation will help us do that:
ln(0.69) = -1.4 × 10^-22 J / (1.38 × 10^-23 J/K * T)
Next, let's isolate T by multiplying both sides of the equation by (1.38 × 10^-23 J/K):
ln(0.69) * (1.38 × 10^-23 J/K) = -1.4 × 10^-22 J / T
Now, divide both sides of the equation by ln(0.69) * (1.38 × 10^-23 J/K):
T = -1.4 × 10^-22 J / (ln(0.69) * (1.38 × 10^-23 J/K))
Calculating this expression will give us the temperature T of the system.