Domain: Combustion spectroscopy

Laser-saturated fluorescence measurements of the hydroxyl radical (OH) are performed in a combustion system. Fluorescence signals corresponding to the P1(11) and P1(16) rovibronic transitions within the electronic transition from X2Pi (v = 0) to A2Sigma (v = 0) are measured to be 11.5 mV and 1.45 mV, respectively.

The spectroscopic parameters for these transitions are as follows:

P1(11):
Wavenumber = 31869.5 cm⁻¹
Einstein A coefficient = 3.54 × 10^5 s⁻¹

P1(16):
Wavenumber = 31492.1 cm⁻¹
Einstein A coefficient = 2.98 × 10^5 s⁻¹

Determine the flame temperature.

For consistency with the two-level laser-saturated fluorescence model, assume that fluorescence detection occurs at the same wavelength as the laser excitation for each spectral line.

Assume the following molecular constants for OH:

Rotational constant: B_e = 18.911 cm⁻¹
Vibration–rotation interaction constant: alpha_e = 0.7242 cm⁻¹.

Grading rubric:
criterion: The solution explicitly uses the saturated two-level fluorescence expression for the lower-state population: n_l^0 = (1 + g_l / g_u) (4πP_d) / (hν_ul A_ul c V_c).
weight: 10
criterion: The solution correctly uses the total rovibronic degeneracy: g = g_el g_rot = (2S + 1)(2J + 1).
weight: 10
criterion: The solution correctly calculates the degeneracies for the P1(11) transition as g_l = 92 and g_u = 42.
weight: 10
criterion: The solution correctly calculates the degeneracies for the P1(16) transition as g_l = 132 and g_u = 62.
weight: 10
criterion: The solution correctly computes the population ratio n_0^i(16) / n_0^i(11) = 0.1487.
weight: 10
criterion: The solution correctly evaluates B_0 = B_e − (1/2)α_e = 18.549 cm⁻¹ within ±0.001 cm⁻¹.
weight: 10
criterion: The final solution uses the provided values B_e = 18.911 cm⁻¹ and α_e = 0.7242 cm⁻¹ to compute the ground-state rotational constant.
weight: 10
criterion: The solution correctly computes F_0(11) = 2414.7 cm⁻¹ and F_0(16) = 4901.8 cm⁻¹, each within ±1%.
weight: 10
criterion: The solution correctly applies n_0^i(16) / n_0^i(11) = (2(16)+1)/(2(11)+1) × exp[−(hc/kT)(F_0(16) − F_0(11))].
weight: 10
criterion: The final answer reports T = 1578 K.
weight: 10

Answer:
1578 K