Question

What is the pressure inside a 10.0 -L flask containing \(14.2 \mathrm{g}\) of \(\mathrm{N}_{2}\) at \(26^{\circ} \mathrm{C} ?\)

Short answer

The pressure inside the 10.0-L flask containing \(14.2\ g\) of \(\mathrm{N}_{2}\) at \(26^{\circ} C\) is approximately \(1.24\ atm\).

Step-by-step solution

To convert the given temperature from Celsius to Kelvin, we use the following formula: \( T(K) = T(^{\circ}C) + 273.15\) For our problem, we have: \(T(K) = 26 + 273.15\) \(T(K) = 299.15\) The temperature is now in Kelvin: \(T = 299.15K\) #Step 2: Determine the number of moles of nitrogen gas#

To determine the number of moles of nitrogen gas, we will use the mass and the molar mass of nitrogen gas: \(n = \frac{mass}{molar\ mass}\) The molar mass of nitrogen gas (\(N_2\)) is \(28.0 \frac{g}{mol}\) (twice the molar mass of a nitrogen atom, since it is diatomic). For our problem, we have: \(n = \frac{14.2\ g}{28.0\ \frac{g}{mol}}\) \(n = 0.507\ mol\) The number of moles is calculated as \(n = 0.507\ mol\) #Step 3: Apply the Ideal Gas Law to find the pressure#

Using the Ideal Gas Law formula, we now have all the necessary information to find the pressure (P): \(PV = nRT\) To find the pressure (P), we will rearrange the formula as follows: \(P = \frac{nRT}{V}\) Now, we plug in the given values: \(P = \frac{(0.507\ mol)(0.0821\ \frac{L \cdot atm}{mol \cdot K})(299.15\ K)}{10.0\ L}\) #Step 4: Calculate the pressure#

Now we can calculate the pressure: \(P = \frac{(0.507)(0.0821)(299.15)}{10.0}\) \(P = 1.24\ atm\) The pressure inside the flask containing nitrogen gas is approximately \(1.24\ atm\).

Key concepts

Pressure Calculation

Understanding pressure calculation in the context of gas laws is pivotal for students studying chemistry. Pressure is one of the fundamental aspects of the Ideal Gas Law, which relates the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas. The pressure of a gas quantifies the force the gas exerts on the walls of its container per unit area.

To find the pressure using the Ideal Gas Law, represented by the formula,
\[ PV = nRT \]
where R is the gas constant, we can rearrange the formula to calculate for pressure if the other variables are known:
\[ P = \frac{nRT}{V} \]

In case studies involving real-life applications, such as determining the pressure of gas within a tire or in this context, a flask, using the Ideal Gas Law provides a straightforward method to do so, given the volume of the gas, the temperature in Kelvin, and the number of moles are known.

Converting Celsius to Kelvin

Temperature is a critical component in the study of gases, as it directly affects gas behavior and properties. The Kelvin scale is the temperature scale commonly used in scientific equations and calculations, including those for gases.

Converting Celsius to Kelvin is a simple yet essential skill for students dealing with temperature-related exercises:
\[ T(K) = T(^{\circ}C) + 273.15 \]

This formula reflects that zero degrees Celsius is equivalent to 273.15 Kelvin, making the Kelvin scale an absolute temperature scale where zero Kelvin is absolute zero, the theoretical point where molecular motion stops. When solving for gas pressures using the Ideal Gas Law, it's vital to convert Celsius temperatures to Kelvin to ensure the correct values are used in calculations.

Molar Mass of Nitrogen

The molar mass of a chemical element or compound is a fundamental concept in chemistry which represents the mass of one mole of its entities (atoms, molecules, ions, etc.). The molar mass of nitrogen (N) is particularly important when dealing with nitrogen gas (N2) since it is a diatomic molecule, meaning two nitrogen atoms are bonded together.

The molar mass of nitrogen is around 14.01 grams per mole (g/mol), but as nitrogen typically exists as diatomic molecules in its gaseous form, the molar mass of nitrogen gas (N2) is roughly double that of a single nitrogen atom:
\[ Molar\ Mass\ of\ N2 = 2 \times 14.01 \frac{g}{mol} \approx 28.02 \frac{g}{mol} \]

This figure is crucial for calculating the number of moles of nitrogen gas using the given mass, as seen in our exercise scenario.

Calculating Moles of Gas

In chemical calculations, particularly those involving gases, it is often necessary to determine the number of moles present. The concept of moles bridges the gap between the microscopic particles we can't see and the macroscopic amounts we can measure. For an ideal gas, the number of moles can be calculated using the mass of the gas and its molar mass with the formula:
\[ n = \frac{mass}{molar\ mass} \]

By using the molar mass of nitrogen gas, for instance, we can calculate the moles of nitrogen in our sample. This step is crucial in utilizing the Ideal Gas Law as it allows us to relate the physical quantities of mass with the volume, temperature, and pressure of a gas. Knowing the correct method to calculate moles enables students to solve a variety of problems involving gas measurements and predictions of gas behavior.