Question

Consider an equimolar mixture of \(\mathrm{CO}_{2}\) and \(\mathrm{O}_{2}\) gases at \(800 \mathrm{~K}\) and a total pressure of \(0.5 \mathrm{~atm}\). For a path length of \(1.2 \mathrm{~m}\), determine the emissivity of the gas.

Short answer

Answer: The emissivity of the equimolar mixture of CO2 and O2 gases at 800 K and a total pressure of 0.5 atm for a path length of 1.2 m is approximately 0.009.

Step-by-step solution

Calculate the partial pressures

Since the mixture is equimolar, equal moles of CO\(_2\) and O\(_2\) are present. Therefore, the partial pressure of each gas can be calculated by dividing the total pressure equally. Total pressure, \(P_{total} = 0.5\) atm Partial pressure of \(\mathrm{CO}_{2}\), \(p_{\mathrm{CO}_{2}} = \frac{1}{2}P_{total}\) Partial pressure of \(\mathrm{O}_{2}\), \(p_{\mathrm{O}_{2}}= \frac{1}{2}P_{total}\)

Use Hottel's correlation to calculate emissivities

Hottel's correlation is an empirical relationship that gives the emissivity of a gas mixture as a function of temperature and partial pressure. For CO\(_2\) and O\(_2\), the formula is: \(E_{mixture} = \left[A_{\mathrm{CO}_{2}}e^{(-B_{\mathrm{CO}_{2}}/p_{\mathrm{CO}_{2}})} + A_{\mathrm{O}_{2}}e^{(-B_{\mathrm{O}_{2}}/p_{\mathrm{O}_{2}})}\right]L\) where \(E_{mixture}\) is the total emissivity of the mixture, \(A_{\mathrm{i}}\) and \(B_{\mathrm{i}}\) are constants for gas i (\(\mathrm{CO}_2\) or \(\mathrm{O}_2\)), \(L\) is the path length, and \(p_{\mathrm{i}}\) is the partial pressure of each gas. The constants for CO\(_2\) and O\(_2\) are: \(A_{\mathrm{CO}_{2}} = 0.035\), \(B_{\mathrm{CO}_{2}} = 300\) \(A_{\mathrm{O}_{2}} = 0.013\), \(B_{\mathrm{O}_{2}} = 360\) We now have the values and can calculate the emissivity of the gas.

Calculate total emissivity

Plug the values of \(A_{\mathrm{i}}\), \(B_{\mathrm{i}}\), partial pressures, and path length into Hottel's correlation formula: \(E_{mixture} = \left[0.035 e^{(-300/(0.5/2))} + 0.013 e^{(-360/(0.5/2))}\right] (1.2)\) \(E_{mixture} \approx 0.009\) So, the emissivity of the equimolar mixture of CO\(_2\) and O\(_2\) gases at 800 K and a total pressure of 0.5 atm for a path length of 1.2 m is approximately 0.009.