Question

At \(0^{\circ} \mathrm{C}\) a \(1.0\)-\(\mathrm{L}\) flask contains \(5.0 \times 10^{-2}\) mole of \(\mathrm{N}_{2}, 1.5 \times\) \(10^{2} \mathrm{mg} \mathrm{O}_{2},\) and \(5.0 \times 10^{21}\) molecules of \(\mathrm{NH}_{3} .\) What is the partial pressure of each gas, and what is the total pressurelin the flask?

Short answer

The partial pressures of the gases are as follows: \(P_{N2} = 1.12\text{ atm}\), \(P_{O2} = 0.105\text{ atm}\), and \(P_{NH3} = 6.32 \times 10^{-3}\text{ atm}\). The total pressure in the flask is \(P_{total} = 1.12 + 0.105 + 6.32 \times 10^{-3} = 1.23\text{ atm}\).

Step-by-step solution

Calculate moles for each gas

We are given the moles for N2 gas, while for the other two gases, we need to convert the given information into moles. For N2: \(5.0 \times 10^{-2}\) moles For O2: We are given the mass of O2 in milligrams. First, let's convert it into grams and then calculate moles. Given mass of O2= 1.5 x 10^2 mg, and 1 g=1000 mg so, Mass of O2 = \(\frac{1.5 \times 10^{2}}{1000}\) g = 0.15 g Moles of O2 = \(\frac{\text{mass of O2}}{\text{molar mass of O2}}\) Moles of O2 = \(\frac{0.15}{32}\) mol For NH3: We are given the number of molecules of NH3. First, let's convert it into moles. Number of molecules of NH3 = \(5.0 \times 10^{21}\) Moles of NH3 = \(\frac{\text{number of molecules}}{\text{Avogadro's number}}\) Moles of NH3 = \(\frac{5.0 \times 10^{21}}{6.022\times10^{23}}\) mol

Calculate partial pressures for each gas

Temperature T = 0°C = 273 K Volume V = 1.0 L Ideal gas constant R = 0.0821 L atm / (K mol) For N2: Using the Ideal Gas Law, \[P_{N2} = \frac{n_{N2} \times R \times T}{V}\] \[P_{N2} = \frac{5.0\times10^{-2} \times 0.0821 \times 273}{1.0}\] For O2: Using the Ideal Gas Law, \[P_{O2} = \frac{n_{O2} \times R \times T}{V}\] \[P_{O2} = \frac{0.15/32 \times 0.0821 \times 273}{1.0}\] For NH3: Using the Ideal Gas Law, \[P_{NH3} = \frac{n_{NH3} \times R \times T}{V}\] \[P_{NH3} = \frac{5.0\times10^{21}/6.022\times10^{23} \times 0.0821 \times 273}{1.0}\]

Calculate the total pressure in the flask

Total pressure inside the flask (P_total) is the sum of the partial pressures of each gas: \[P_{total} = P_{N2} + P_{O2} + P_{NH3}\] Calculate the total pressure using the calculated partial pressures for each gas.

Key concepts

Ideal Gas Law

The ideal gas law is an essential equation in chemistry that allows you to relate the pressure, volume, temperature, and number of moles of a gas. It is expressed as \(PV = nRT\), where:

• \(P\) stands for pressure, measured often in atmospheres (atm).


• \(V\) is the volume of the gas, often given in liters (L).


• \(n\) represents the number of moles, which connects the mass and type of substance to its quantity.


• \(R\) is the ideal gas constant, commonly used as \(0.0821\, \text{L atm K}^{-1} \text{mol}^{-1}\).


• \(T\) is the temperature in Kelvin.

To apply the ideal gas law, ensure that the temperature is in Kelvin, as this standardizes the calculations. You can convert Celsius to Kelvin by adding 273 to the Celsius temperature. This understanding will help you compute the partial pressures of gases, such as nitrogen (\(N_2\)), oxygen (\(O_2\)), and ammonia (\(NH_3\)) in a mixture, by substituting moles and temperature values into the equation.

Mole Calculations

Moles are a fundamental concept in chemistry, serving as a bridge between the microscopic world of atoms and molecules and the macroscopic quantities we can measure. A mole represents \(6.022 \times 10^{23}\) entities (atoms, molecules, etc.).

• For gases like \(N_2\), if the moles are given directly, simply use these in your calculations.
• For substances like \(O_2\), where the mass might be provided, you must convert mass to moles using the formula: \(\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\).


• Molar mass is available on the periodic table. For example, \(O_2\) has a molar mass of \(32\, \text{g/mol}\), so you divide the given mass by \(32\).

Clear calculations allow you to use the ideal gas law effectively by transforming various units of measurement into moles, ensuring your results are accurate. Merging this knowledge improves your ability to calculate how gases will behave under different conditions.

Avogadro's Number

Avogadro’s number, \(6.022 \times 10^{23}\), is the key to understanding the quantity in a mole of substance, linking the microscopic to the macroscopic. It represents the number of atoms or molecules in one mole.

• For example, with \(NH_3\), if you know the number of molecules, you can find the moles by dividing by Avogadro's number: \(\text{moles} = \frac{\text{number of molecules}}{6.022 \times 10^{23}}\).


• This conversion helps equate the microscopic count to a macroscopic measure (moles), facilitating the use of the ideal gas law to find properties like pressure.


• By understanding Avogadro's number, you can transition from theoretical molecule counts to practical measurements needed in computations and experiments.

Embracing Avogadro's number allows you to grasp the scope of chemical reactions and transformations, ultimately refining your ability to solve chemistry problems effectively.