Question

The density of a noble gas is \(2.71 \mathrm{~g} / \mathrm{L}\) at \(3.00 \mathrm{~atm}\) and \(0^{\circ} \mathrm{C}\). Identify the gas.

Short answer

The gas is Neon (Ne).

Step-by-step solution

- Identify the given variables

The problem provides the density of the gas as \(2.71 \, \mathrm{g/L}\), the pressure as \(3.00 \, \mathrm{atm}\), and the temperature as \(0^{\circ} \mathrm{C} = 273 \mathrm{~K}\).

- Use the Ideal Gas Law

The Ideal Gas Law is given by \(PV = nRT\). Rearrange this equation to solve for the molar mass (M): \[M = \frac{{dRT}}{{P}}\] where \(d\) is the density, \(R\) is the gas constant (0.0821 \(\mathrm{L} \cdot \mathrm{atm} / (\mathrm{mol} \cdot \mathrm{K})\)), \(T\) is the temperature in Kelvin, and \(P\) is the pressure.

- Substitute the given values

Substitute the values into the formula: \[M = \frac{{2.71 \, \mathrm{g/L} \times 0.0821 \, \mathrm{L \cdot atm/(mol \cdot K)} \times 273 \, \mathrm{K}}}{{3.00 \, \mathrm{atm}}} \]

- Calculate the molar mass

Perform the calculations: \[M = \frac{{2.71 \times 0.0821 \times 273}}{{3.00}} = \frac{{60.90873}}{{3.00}} \approx 20.30 \, \mathrm{g/mol} \]

- Identify the noble gas

Compare the calculated molar mass to the known molar masses of noble gases. The calculated molar mass of approximately \(20.30 \, \mathrm{g/mol} \) is closest to the molar mass of neon (Ne), which is \(20.18 \, \mathrm{g/mol}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is essential for understanding how gases behave under different conditions. The formula for the Ideal Gas Law is \(PV=nRT\). In this equation:

• \(P\) stands for pressure.
• \(V\) represents volume.
• \(n\) is the number of moles of the gas.
• \(R\) is the gas constant, which is \(0.0821 \, \mathrm{L \, atm/(mol \, K)}\).
• \(T\) is the temperature in Kelvin.

By rearranging this equation, you can solve for different variables, like molar mass. This adaptability makes the Ideal Gas Law very useful for different types of gas-related problems.

Gas Density

Gas density is a measure of how much mass a gas has in a given volume. It is usually expressed in \(\mathrm{g/L}\). In the exercise, the density of the gas is given as \(2.71 \, \mathrm{g/L}\). Density can be linked to other properties of gases through the Ideal Gas Law. For instance, knowing the density allows us to calculate the molar mass if we also know the pressure and temperature. This relationship is key for identifying the type of gas you're dealing with by using the rearranged Ideal Gas Law formula.

Molar Mass Calculation

Calculating the molar mass of a gas involves using the rearranged Ideal Gas Law formula \(M = \frac{dRT}{P}\). Here:

• \(M\) is the molar mass.
• \(d\) is density.
• \(R\) is the gas constant.
• \(T\) is temperature in Kelvin (273 K for \(0^{\circ}\, \mathrm{C}\)).
• \(P\) is pressure.

By substituting the values we know into the formula, we solve for \(\mathrm{M}\). For example, in the given exercise:
\[M = \frac{2.71 \, \mathrm{g/L} \times 0.0821 \mathrm{L \, atm/(mol \, K)} \times 273 \, \mathrm{K}}{3.00 \, \mathrm{atm}} = \approx 20.30 \, \mathrm{g/mol}\]. This calculation matches closely with the molar mass of neon, allowing us to identify the gas.

Noble Gases

Noble gases are a group of chemical elements with similar properties. They are all odorless, colorless, monatomic gases with very low chemical reactivity. The noble gases include:

• Helium \(\mathrm{He}\)
• Neon \(\mathrm{Ne}\)
• Argon \(\mathrm{Ar}\)
• Krypton \(\mathrm{Kr}\)
• Xenon \(\mathrm{Xe}\)
• Radon \(\mathrm{Rn}\)

Each of these gases has a different molar mass, enabling us to identify them by calculating the molar mass from density, pressure, and temperature. For example, Neon (Ne) has a molar mass of approximately \(20.18 \, \mathrm{g/mol}\), which closely matches the solution of the given exercise.