Question

What is the pressure inside a 10.0 -L flask containing \(14.2 \mathrm{~g}\) of \(\mathrm{N}_{2}\) at \(\begin{array}{ll}26 & \mathrm{C} \\ ?\end{array}\)

Short answer

The pressure inside the 10.0-L flask containing 14.2 g of Nitrogen gas at \(26^{\circ}C\) (\(299.15 K\)) is approximately 1.245 atm.

Step-by-step solution

Conversion of mass to moles

The molar mass of Nitrogen gas (\(N_2\)) is 28.0 g/mol (14.0 g/mol * 2, since N2 contains two Nitrogen atoms). To convert the mass of the Nitrogen gas to moles, use the following formula: \(n = \frac{m}{M}\) Where: - n = Number of moles - m = Mass of Nitrogen gas = 14.2 g - M = Molar mass of Nitrogen gas = 28.0 g/mol \(n = \frac{14.2 \, \text{g}}{28.0 \, \text{g/mol}}\)

Calculation of moles

Calculating the number of moles of Nitrogen gas: \(n = \frac{14.2 \, \text{g}}{28.0 \, \text{g/mol}}\) \(n = 0.5071 \, \text{mol}\)

Conversion of Celsius to Kelvin

As the Ideal Gas Law requires the temperature to be in Kelvin (K), we need to convert the given temperature from Celsius (°C) to Kelvin (K). The formula to do this is: \(T_{K} = T_{C} + 273.15\) Where: - \(T_{K}\) = Temperature in Kelvin (K) - \(T_{C}\) = Temperature in Celsius (°C) \(T_{K} = 26°C + 273.15\)

Calculation of Temperature in Kelvin

Calculating the temperature in Kelvin: \(T_{K} = 26°C + 273.15\) \(T_{K} = 299.15 \, K\)

Ideal Gas Law calculation

Now, using the Ideal Gas Law, we can find the pressure inside the 10.0 -L flask: \(PV = nRT\) where - \(P\) = pressure - \(V\) = volume = 10.0 L - \(n\) = 0.5071 mol - \(R\) = 0.08206 L.atm.K-1.mol-1 - \(T\) = 299.15 K Solving for \(P\): \(P = \frac{nRT}{V} \) \(P = \frac{0.5071 \, \text{mol} \cdot 0.08206 \, \frac{\text{L.atm}}{\text{K.mol}} \cdot 299.15 \, K}{10.0 \, \text{L}}\)

Calculation of Pressure

Calculating the pressure inside the flask: \(P = \frac{0.5071 \, \text{mol} \cdot 0.08206 \, \frac{\text{L.atm}}{\text{K.mol}} \cdot 299.15 \, K}{10.0 \, \text{L}}\) \(P = 1.245 \, \text{atm}\) The pressure inside the 10.0-L flask containing 14.2 g of Nitrogen gas at 26°C is approximately 1.245 atm.

Key concepts

Molar Mass Calculation

Understanding how to calculate molar mass is essential for a wide range of problems in chemistry, including the usage of the Ideal Gas Law. The molar mass is the weight of one mole (6.022 \( \times \) 10^23 particles) of a substance and is expressed in grams per mole (g/mol). For a molecular compound like nitrogen gas \( (N_2) \), you calculate the molar mass by summing the molar mass of each atom within the molecule.

For \( N_2 \), with each nitrogen (N) atom having an atomic mass of approximately 14.0 g/mol, the molar mass becomes:
\[ Molar \text{ } Mass \text{ } of \text{ } N_2 = 2 \times 14.0 \text{ } g/mol = 28.0 \text{ } g/mol \]
We use this molar mass as a conversion factor to relate mass in grams to the number of moles. This step is crucial because the Ideal Gas Law involves moles, not grams, and therefore requires the conversion from mass to moles, using the formula:\[ n = \frac{m}{M} \]where \( n \) is the number of moles, \( m \) is the mass, and \( M \) is the molar mass of the gas.

Temperature Conversion

Temperature plays a vital role in the behavior of gases, and in the context of the Ideal Gas Law, it’s important to use the correct temperature unit. The Ideal Gas Law requires temperature to be in Kelvin (K), which is an absolute temperature scale. This is because 0 K, also known as absolute zero, represents the point where particles have minimal thermal motion.

To convert Celsius to Kelvin, which is a necessary step before applying the Ideal Gas Law, use the formula:\[ T_K = T_C + 273.15 \]
Here, \( T_K \) is the temperature in Kelvin and \( T_C \) is the temperature in Celsius. This conversion is important because temperature in Celsius starts at the freezing point of water (0°C) and is not appropriate for gas law calculations that require an absolute scale. For instance, in our exercise, the temperature given is 26°C. After converting to Kelvin, we get:\[ T_K = 26°C + 273.15 = 299.15 K \]
Now, the temperature is in the correct form to be utilized within the Ideal Gas Law equation.

Pressure Calculation

The concept of pressure is at the heart of many gas-related calculations. In the context of the Ideal Gas Law, pressure represents the force that the gas particles exert on the walls of their container per unit area. It is essential to find this pressure to understand the behavior of the gas under various conditions.

The Ideal Gas Law equation \( PV = nRT \) incorporates pressure \( (P) \), volume \( (V) \), number of moles of the gas \( (n) \), the universal gas constant \( (R) \), and temperature \( (T) \). When calculating pressure, we rearrange the equation to:\[ P = \frac{nRT}{V} \]
In our example, once we have converted the mass of nitrogen to moles and the Celsius temperature to Kelvin, we can insert those values, along with the volume of the flask and the universal gas constant, into this formula to compute the pressure:
\[ P = \frac{0.5071 \text{ } mol \times 0.08206\text{ } \frac{L.atm}{K.mol} \times 299.15\text{ } K}{10.0\text{ } L} = 1.245\text{ } atm \]
This calculation shows that the pressure inside the 10.0-L flask containing 14.2 g of Nitrogen gas at 26°C is approximately 1.245 atm.