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The Ideal Gas Law is a fundamental principle in chemistry to relate different properties of gases. It provides a simple yet powerful relation between pressure, volume, temperature, and number of moles of a gas. The formula is expressed as:\[PV = nRT\]where:- \(P\) is the pressure of the gas,- \(V\) is the volume it occupies,- \(n\) is the number of moles,- \(R\) is the Universal Gas Constant, which is \(0.0821\, \text{L}\cdot\text{atm/mol}\cdot\text{K}\),- \(T\) is the temperature in Kelvin.To find the partial pressure of a single gas in a mixture using the Ideal Gas Law, we rearrange the formula to isolate pressure:\[P = \frac{nRT}{V}\]Make sure to convert the temperature to Kelvin by adding 273.15 to the Celsius measurement before using this equation. This conversion is crucial because the Kelvin scale is defined in such a way that it avoids negative values, simplifying calculations of thermodynamic properties.

Understanding how to convert masses into moles is essential when working with gases. Moles represent a quantity that corresponds to Avogadro's number \(6.022 \times 10^{23}\) of molecules or atoms. The formula to convert mass into moles is:\[n = \frac{\text{Mass in grams}}{\text{Molar mass in g/mol}}\]Let's break this down with the gases from our exercise:

• Hydrogen (\(\mathrm{H}_{2}\)): Its molar mass is approximately 2.02 g/mol. To convert 0.513 grams of hydrogen into moles, divide the mass by its molar mass: \(n_{\mathrm{H}_{2}} = \frac{0.513}{2.02} \approx 0.254\text{ mol}\).


• Nitrogen (\(\mathrm{N}_{2}\)): With a molar mass of approximately 28.02 g/mol, you can find moles of nitrogen similarly: \(n_{\mathrm{N}_{2}} = \frac{16.1}{28.02} \approx 0.575\text{ mol}\).

Converting mass to moles facilitates using the ideal gas law, allowing us to predict how the gas behaves under different physical conditions.

In a gas mixture, the total pressure is the sum of partial pressures of its individual gases. This property is due to Dalton's Law of Partial Pressures, which states that each gas in a mixture exerts pressure independently. Hence, the pressure of each component is determined as if it alone occupied the volume:

• Partial pressure of \(\mathrm{H}_{2}\): Using the ideal gas law, this is calculated as \(P_{\mathrm{H}_{2}} = \frac{(0.254\,\text{mol})(0.0821\,\text{L}\cdot\text{atm/mol}\cdot\text{K})(293.15\,\text{K})}{10.0\,\text{L}} \approx 0.610\,\text{atm}\).


• Partial pressure of \(\mathrm{N}_{2}\): Calculated similarly to \(\mathrm{H}_{2}\), it is \(P_{\mathrm{N}_{2}} = \frac{(0.575\,\text{mol})(0.0821\,\text{L}\cdot\text{atm/mol}\cdot\text{K})(293.15\,\text{K})}{10.0\,\text{L}} \approx 1.38\,\text{atm}\).

By summing the partial pressures, we can obtain the total pressure of the mixture. Understanding how to calculate the partial pressures is key to analyzing the behavior of gas mixtures in different chemical reactions and processes.