Question

A 900.0-mL flask contains \(1.16 \mathrm{mg} \mathrm{O}_{2}\) and \(0.42 \mathrm{mg}\) He at \(15^{\circ} \mathrm{C}\). Calculate the partial pressures of oxygen and of helium in the flask. What is the total pressure?

Short answer

The partial pressures are 0.000955 atm for oxygen and 0.002778 atm for helium; total pressure is 0.003733 atm.

Step-by-step solution

Convert Mass to Moles for Oxygen

First, convert the mass of oxygen into moles using the molar mass of oxygen gas, which is approximately 32.00 g/mol. The given mass is 1.16 mg, or 0.00116 g.\[\text{Moles of } \mathrm{O_2} = \frac{0.00116 \text{ g}}{32.00 \text{ g/mol}} = 3.625 \times 10^{-5} \text{ moles}\]

Convert Mass to Moles for Helium

Next, convert the mass of helium into moles using the molar mass of helium, which is approximately 4.00 g/mol. The given mass is 0.42 mg, or 0.00042 g.\[\text{Moles of He} = \frac{0.00042 \text{ g}}{4.00 \text{ g/mol}} = 1.05 \times 10^{-4} \text{ moles}\]

Calculate the Volume in Liters

Convert the volume from milliliters to liters, knowing that there are 1000 milliliters in a liter.\[V = \frac{900.0 \text{ mL}}{1000} = 0.900 \text{ L}\]

Use Ideal Gas Law for Partial Pressure of Oxygen

Use the ideal gas law to find the partial pressure of oxygen. The ideal gas law is \(PV = nRT\). Use \(R = 0.0821 \) L atm/(mol K), and convert temperature to Kelvin: \( T = 15 + 273 = 288 \) K.\[P_{\mathrm{O_2}} = \frac{(3.625 \times 10^{-5}) \cdot 0.0821 \cdot 288}{0.900} = 0.000955 \text{ atm}\]

Use Ideal Gas Law for Partial Pressure of Helium

Apply the ideal gas law to find the partial pressure of helium under the same conditions.\[P_{\mathrm{He}} = \frac{(1.05 \times 10^{-4}) \cdot 0.0821 \cdot 288}{0.900} = 0.002778 \text{ atm}\]

Calculate Total Pressure

To find the total pressure in the flask, add the partial pressures of oxygen and helium together.\[P_{\text{total}} = P_{\mathrm{O_2}} + P_{\mathrm{He}} = 0.000955 + 0.002778 = 0.003733 \text{ atm}\]

Key concepts

Partial Pressure Calculation

Partial pressure refers to the pressure that a single gas in a mixture would exert if it occupied the entire volume on its own. In a mixture of gases, like oxygen (O₂) and helium (He), each gas contributes to the total pressure based on its amount and identity. The ideal gas law, represented as \(PV = nRT\), is helpful here because it relates pressure (P), volume (V), number of moles (n), the ideal gas constant (R), and temperature (T).

In the context of calculating partial pressures, you would apply the ideal gas law individually to each gas. For example, when calculating the partial pressure of oxygen in the given flask, you would plug the number of moles for oxygen, the volume of the flask, the temperature in Kelvin, and the ideal gas constant into the ideal gas equation to solve for pressure. The same process applies individually for helium.

• Calculate the number of moles for each gas.
• Use the ideal gas law: \( P = \frac{nRT}{V} \).
• Fill into the equation with the correct values to find each gas's partial pressure.

Mole Conversion

Mole conversion is a crucial step in determining quantities of substances in many chemistry problems. Here, we have to convert mass in milligrams to moles for gases such as oxygen and helium, using their molar mass.

The conversion formula is:
\[ \ ext{Moles} = \rac{\text{Mass in grams}}{\text{Molar Mass}} \]

• The mass of each gas should be converted from milligrams to grams first (1 mg = 0.001 g).
• Then, use the molar mass of each gas to find the number of moles:
• For oxygen: divide by 32.00 g/mol.
• For helium: divide by 4.00 g/mol.

By converting to moles, we make it possible to apply the ideal gas law accurately. This step ensures you're working with the correct amount of substance according to chemical standards.

Total Pressure

Total pressure in a system with multiple gases is simply the sum of the partial pressures of all gases present. This concept is derived from Dalton's Law of Partial Pressures, which states that in a mixture of non-reacting gases, each gas contributes to the total pressure independently.

To find the total pressure, you follow these steps:

• After calculating each gas's partial pressure separately using the ideal gas law, simply add them together.
• The total pressure is reflective of the combined effect of both gases filling the volume of the flask.
• The formula is straightforward: \( P_{\text{total}} = P_{\text{O}_2} + P_{\text{He}} \).

Understanding total pressure is important because it relates back to how gases behave when mixed, allowing prediction and control of gas behaviors in various applications.

Oxygen and Helium Gases

Oxygen and helium are two distinct gases often used in chemical and industrial applications. In this exercise, understanding their individual properties is key to calculating their contributions to the total pressure in the flask.

**Important Characteristics:**

• **Oxygen (O₂):** A diatomic molecule with a molar mass of 32.00 g/mol. It is essential for respiration and combustion processes.
• **Helium (He):** A noble gas with a molar mass of 4.00 g/mol. It is very light, chemically inert, and used in applications where non-reactivity is crucial.

When mixed in a flask, their differences in molar mass and density affect their partial pressures. Despite having different chemical properties, applying the ideal gas law allows us to simplify calculations for mixtures involving these gases.