Question

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

Short answer

Question: Calculate the molar concentration of an equimolar mixture of CO2 and O2 gases, given the following conditions: total pressure of 0.5 atm, temperature of 800 K, and a path length proportional to a volume of 1.2 m3. Answer: The molar concentration for each gas (CO2 and O2) is 4.573 mol.

Step-by-step solution

Calculate the molar concentration of each gas

Since the mixture is equimolar, the molar concentration of each gas is equal. Let's first calculate the total moles of gas within the given total pressure. We can calculate the moles of gas using the ideal gas law: \(PV = nRT\) Where: \(P = 0.5 \ \text{atm}\) \(V = 1.2 \ \text{m}^3\) (Assuming the path length is proportional to volume) \(T = 800 \ \text{K}\) \(R = 0.0821 \ \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}}\) First, we need to convert the volume to liters: \(V (L) = 1.2 \ \text{m}^3 \times \frac{1000 \ \text{L}}{1 \ \text{m}^3} = 1200 \ \text{L}\) Now, we can solve for \(n\) (the total moles): \(n = \frac{PV}{RT} = \frac{(0.5 \ \text{atm})(1200 \ \text{L})}{(0.0821 \ \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}})(800 \ \text{K})} = 9.145 \ \text{mol}\) The molar concentration for each gas (n') is half of the total moles: \(n' = \frac{9.145 \ \text{mol}}{2} = 4.573 \ \text{mol}\)

Calculate partial pressures of each gas

Now we need to find the partial pressure of each gas, using the mole fraction: For an equimolar mixture, the mole fraction is equal to \(\frac{1}{2}\) for each gas. Partial pressure\( \ (P_{i}) = \text{mole fraction}_i \times P_{\text{total}}\) For both \(\mathrm{CO}_{2}\) and \(\mathrm{O}_{2}\) gases: \(P_{\mathrm{CO}_{2}} = P_{\mathrm{O}_{2}} = \frac{1}{2} \times 0.5 \ \text{atm} = 0.25\) atm

Determine the weighted sum of emissivities

Now, we need to determine the individual emissivities for each gas \((\varepsilon_i)\) and combine them into a single emissivity value for the mixture using the weighted sum of the emissivities: \(\varepsilon_\text{total} = \frac{\sum_{i=1}^N \varepsilon_i P_i}{\sum_{i=1}^N P_i}\) Where \(P_i\) is the partial pressure of each gas and \(N\) is the number of gases in the mixture. Unfortunately, we cannot compute the individual emissivities because the values are not provided, and these values are based on experimental measurements. Typically, one would have to consult a table or database to find the corresponding emissivity values for each gas at the given conditions (temperature, pressure, and path length). If we had the emissivities for each gas, we would substitute them in the above equation and calculate the result. However, this is beyond the scope of the current exercise, as determining these values requires specific experimental data.

Key concepts

Equimolar Mixture

An equimolar mixture occurs when two or more components in a mixture have equal numbers of moles. This means that the quantity of each substance in terms of moles is the same. For the gas mixture in the exercise, we have a combination of carbon dioxide \((\mathrm{CO_2})\) and oxygen \((\mathrm{O_2})\). This equal-mole condition simplifies calculations since the mole fraction for each component will be the same.
Understanding equimolar mixtures helps in calculating partial pressures and molar concentrations, as it removes the need to know the absolute number of moles of each component individually. It simply becomes a factor of dividing the total moles by the number of components. In our example, there are two gases, so each would have half of the total moles.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physical chemistry and physics. It links pressure \(P\), volume \(V\), temperature \(T\), and number of moles \(n\) of a gas, characterized by the equation:
\[ PV = nRT \] Here, \(R\) is the ideal gas constant. This law assumes that the gas is "ideal," meaning that the molecules occupy no space and experience no intermolecular attractions.
For our exercise, the Ideal Gas Law helps us calculate the total number of moles in the gas mixture. By rearranging the formula to solve for \(n\), we can find the total moles in the volume specified. Remember, Unit conversion is key - here, 1 cubic meter was converted to liters for consistency with \(R\)'s units.
This law is crucial for anyone studying gaseous systems as it provides a comprehensive model for determining one property of the gas if the others are known.

Partial Pressure

The concept of partial pressure is important for understanding gas mixtures. It refers to the pressure that a single gas in a mixture would exert if it were alone. This is useful in calculating total pressures and compositions of gas mixtures.
Partial pressures can be determined using the mole fraction of each gas in the mixture and the total pressure, following the equation:
\[ P_i = \text{mole fraction}_i \times P_{\text{total}} \]
For equimolar mixtures, like the one in our problem, the mole fraction is equally divided among the components. Thus, each gas in the mixed system has a partial pressure that is a fraction of the total pressure.
Understanding partial pressures is essential in fields like chemical engineering and environmental science, where predicting gas behavior under different conditions is crucial.

Molar Concentration

Molar concentration, also known as molarity, is a way of expressing the concentration of a species in a mixture. This is usually in units of moles per liter. In gas mixtures, molar concentration can be tied back to partial pressures and total moles.
In the context of an equimolar mixture, determining the molar concentration means dividing the total moles of the gas by its volume. Since our conditions are known, calculating molar concentration becomes straightforward:
- First, determine the total moles with Ideal Gas Law - Then, find moles per component (here it would be a straightforward half)
Understanding molar concentrations helps in determining other properties, like density and reactivity, of the mixture.