Question

Calculate the ratio of effusion rates for nitrogen $\left({\mathrm{N}}_2\right)$ and neon (Ne).

Short answer

The ratio of effusion rates for nitrogen (${\mathrm{N}}_2$) and neon (Ne) is approximately 0.845, meaning nitrogen gas effuses about 0.845 times faster than neon gas.

Step-by-step solution

Find the molar mass of nitrogen and neon

To calculate the molar mass, we use the atomic mass of each element. In the periodic table, the atomic mass of nitrogen (N) is approximately 14 u and neon (Ne) is approximately 20 u. Since nitrogen gas (${\mathrm{N}}_2$) consists of two nitrogen atoms, the molar mass of ${\mathrm{N}}_2$ is: M(${\mathrm{N}}_2$) = Atomic mass of N * 2 The molar mass of neon gas (Ne) is the same as its atomic mass: M(Ne) = Atomic mass of Ne Now we can calculate the molar mass for both gases: M(${\mathrm{N}}_2$) = 14 * 2 = 28 g/mol M(Ne) = 20 g/mol

Apply Graham's Law of Effusion

Graham's Law of Effusion relates the effusion rates of two gases to the inverse ratio of the square roots of their molar masses. Mathematically, the formula for Graham's Law of Effusion is: $\frac{Rat{e}_1}{Rat{e}_2}=\sqrt{\frac{M_2}{M_1}}$ In our case, we want to find the ratio of effusion rates for nitrogen (${\mathrm{N}}_2$) and neon (Ne). Therefore, we will substitute Rate 1 with Rate(${\mathrm{N}}_2$) and Rate 2 with Rate(Ne), and their respective molar masses: $\frac{Rat{e}_{({\mathrm{N}}_2)}}{Rat{e}_{(Ne)}}=\sqrt{\frac{M_{(Ne)}}{M_{({\mathrm{N}}_2)}}}$

Calculate the ratio of effusion rates

We will now plug in the molar masses that we calculated in step 1 into the formula: $\frac{Rat{e}_{({\mathrm{N}}_2)}}{Rat{e}_{(Ne)}}=\sqrt{\frac{20}{28}}$ To simplify the expression, we can calculate the square root: $\frac{Rat{e}_{({\mathrm{N}}_2)}}{Rat{e}_{(Ne)}}\approx \sqrt{0.714}$ $\frac{Rat{e}_{({\mathrm{N}}_2)}}{Rat{e}_{(Ne)}}\approx 0.845$ The ratio of effusion rates for nitrogen to neon is approximately 0.845. It means nitrogen gas effuses about 0.845 times faster than neon gas.

Key concepts

Molar Mass Calculation

Calculating the molar mass of a gas is an important step in determining its effusion rate. The molar mass represents the mass of one mole of a substance. In this context, it's measured in grams per mole (g/mol). For nitrogen gas, which is diatomic ( _2), we start by finding the atomic mass of a single nitrogen atom. From the periodic table, that's approximately 14 atomic mass units (u). Since nitrogen gas contains two atoms, the molar mass of nitrogen _2 can be calculated as:

• Molar Mass of _2 = 14 u * 2 = 28 g/mol

For neon, which is monatomic, the calculation is simpler. Neon has an atomic mass of 20 u, so its molar mass is:

• Molar Mass of Ne = 20 g/mol

Understanding molar mass is crucial in comparing different gases' rates of effusion.

Effusion Rates

Effusion is the process by which gas particles escape from a container through a small opening. Graham's Law of Effusion helps us compare the rates at which two gases effuse. It states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it is expressed as:

• $\frac{Rat{e}_1}{Rat{e}_2}=\sqrt{\frac{M_2}{M_1}}$

"Rate" refers to how fast a gas escapes, and the "M" stands for molar mass. This formula allows us to compare the effusion rate of one gas to another using their molar masses. Larger molar masses translate to slower effusion rates, and vice versa. This relationship is essential for predicting how gases behave in different conditions.

Nitrogen Gas

Nitrogen gas (${\mathrm{N}}_2$) is a diatomic molecule, meaning it consists of two nitrogen atoms. As one of the most abundant gases in Earth's atmosphere, nitrogen is colorless, odorless, and largely inert. Its properties make it a good candidate for understanding effusion processes, as it won't react with other elements during experimentation.In terms of effusion, nitrogen is used frequently in studies because of its simplicity and availability. Calculating the effusion rate of nitrogen gas using Graham's Law requires knowledge of its molar mass, which we found earlier to be 28 g/mol. This molar mass is crucial for comparing nitrogen's effusion rate to other gases, like neon, by plugging it into Graham's equation.

Neon Gas

Neon gas (Ne) is a noble gas, known for its stability and lack of reaction with other elements. As a monatomic gas, it consists of single neon atoms rather than molecules. With a molar mass of 20 g/mol, neon is lighter than many other gases, which influences how quickly it effuses. Neon's notable property in this context is how its effusion rate can be directly calculated against other gases, such as nitrogen, using Graham’s Law. By understanding that neon's lighter molar mass compared to nitrogen makes it effuse more slowly, students can grasp the fundamental concepts of effusion. This is a key insight when studying practical applications like gas separation and purification processes.