To rank the gases from most dense to least dense at 1.00 atm and 298 K, we can use the ideal gas law and the molar mass of each gas. The density (\(d\)) of a gas can be calculated using the formula:
\[ d = \frac{PM}{RT} \]
where:
- \(P\) is the pressure (1.00 atm),
- \(M\) is the molar mass of the gas (in g/mol),
- \(R\) is the ideal gas constant (0.0821 L·atm/(K·mol)),
- \(T\) is the temperature (298 K).
Using this formula, we can determine the density of each gas given their molar masses.
Here’s a general approach to rank the gases based on their molar masses:
- Calculate or look up the molar mass for each gas.
- Use the density formula to find the density at the given conditions.
- Rank the gases based on the calculated densities.
Let's consider the following example gases:
- Carbon Dioxide (CO₂) - Molar mass ~ 44.01 g/mol
- Oxygen (O₂) - Molar mass ~ 32.00 g/mol
- Nitrogen (N₂) - Molar mass ~ 28.02 g/mol
- Hydrogen (H₂) - Molar mass ~ 2.02 g/mol
Calculating the Densities
-
CO₂: \[ d_{CO_2} = \frac{(1.00 , \text{atm})(44.01 , \text{g/mol})}{(0.0821 , \text{L·atm/(K·mol)})(298 , \text{K})} \approx 1.98 , \text{g/L} \]
-
O₂: \[ d_{O_2} = \frac{(1.00 , \text{atm})(32.00 , \text{g/mol})}{(0.0821 , \text{L·atm/(K·mol)})(298 , \text{K})} \approx 1.43 , \text{g/L} \]
-
N₂: \[ d_{N_2} = \frac{(1.00 , \text{atm})(28.02 , \text{g/mol})}{(0.0821 , \text{L·atm/(K·mol)})(298 , \text{K})} \approx 1.25 , \text{g/L} \]
-
H₂: \[ d_{H_2} = \frac{(1.00 , \text{atm})(2.02 , \text{g/mol})}{(0.0821 , \text{L·atm/(K·mol)})(298 , \text{K})} \approx 0.09 , \text{g/L} \]
Ranking the Gases
After calculating the densities, we can rank the gases from most dense to least dense as follows:
- CO₂ (1.98 g/L)
- O₂ (1.43 g/L)
- N₂ (1.25 g/L)
- H₂ (0.09 g/L)
Thus, the final ranking from most dense to least dense is:
CO₂ > O₂ > N₂ > H₂
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