Question

At what temperature would \(4.25 \mathrm{~g}\) of oxygen gas, \(\mathrm{O}_{2}\), exert a pressure of \(784 \mathrm{~mm} \mathrm{Hg}\) in a 2.51 -L container?

Short answer

The temperature at which \(4.25 \mathrm{~g}\) of oxygen gas would exert a pressure of \(784 \mathrm{~mm \space Hg}\) in a 2.51-L container is approximately \(243.86 \mathrm{~K}\).

Step-by-step solution

Convert the given pressure from mm Hg to atm

To convert 784 mm Hg to atm, use the following conversion factor: 1 atm = 760 mm Hg. So, \(P(atm) = \frac{784 \mathrm{~mm \space Hg}}{760 \mathrm{~mm \space Hg / atm}} = 1.0316 \mathrm{~atm}\).

Convert the mass of oxygen gas to moles

To convert the mass of oxygen gas (\(4.25 \mathrm{~g}\)) to moles, we will need the molar mass of oxygen gas. An oxygen molecule consists of two oxygen atoms, and the atomic mass of an oxygen atom is approximately 16 g/mol. So, the molar mass of oxygen gas is 32 g/mol. Now, we can find the number of moles using: \(n = \frac{mass}{molar \space mass}\) \(n = \frac{4.25 \mathrm{~g}}{{32 \mathrm{~g/mol}}} = 0.1328 \mathrm{~mol}\).

Calculate the temperature using the Ideal Gas Law

We have the pressure (P), volume (V), number of moles (n), and gas constant (R). Now we can use the Ideal Gas Law equation to calculate the temperature (T): \(PV = nRT\) Substitute the values and solve for T: \(T = \frac{PV}{nR} = \frac{1.0316 \mathrm{~atm} \times 2.51 \mathrm{~L}}{0.1328 \mathrm{~mol} \times 0.0821 \mathrm{~L \cdot atm \cdot K^{-1} \cdot mol^{-1}}}\) \(T \approx 243.86 \mathrm{~K}\) The temperature at which \(4.25 \mathrm{~g}\) of oxygen gas would exert a pressure of \(784 \mathrm{~mm \space Hg}\) in a 2.51-L container is approximately 243.86 K.

Key concepts

Gas Pressure Conversion

Converting gas pressure from one unit to another is a foundational concept in chemistry. It is essential to understand this conversion to solve problems related to the Ideal Gas Law. - **Common Pressure Units:** The most commonly used units for gas pressure include atmospheres (atm), millimeters of mercury (mm Hg), pascals, and barrs. Each of these represents the force exerted by a gas over a specific area. - **Conversion Factors:** It’s important to use correct conversion factors between units. For instance, 1 atm is equivalent to 760 mm Hg. This conversion factor is essential when converting between these units. In our exercise, the pressure was provided in mm Hg, which needed conversion to atm to apply in the Ideal Gas Law. The conversion was done by dividing the given pressure by 760 mm Hg/atm, resulting in 1.0316 atm. This conversion ensures that we can use the gas law equation accurately in calculations, as many formulas, including the Ideal Gas Law, use pressure in units of atm.

Molar Mass Calculation

Finding the molar mass of a gas involves determining the mass of one mole of that substance. This is particularly important when dealing with molecular gases like oxygen. - **Molecular Composition:** Oxygen gas (O_2) is composed of two oxygen atoms. Knowing the atomic mass of an element helps us calculate the molar mass of more complex molecules. Each oxygen atom has an atomic mass approximately 16 g/mol. Thus, the total for O_2 is 32 g/mol. - **Purpose of Molar Mass Calculation:** This value allows conversion of the mass of a substance to the number of moles, a basic unit in chemistry relating quantities of matter to particle numbers. In our example, the molar mass calculation was used to convert 4.25 grams of oxygen gas to moles (0.1328 mol) using the equation \( n = \frac{mass}{molar \space mass} \). This step is crucial for using the Ideal Gas Law, as the law requires the number of moles to relate pressure, volume, and temperature effectively.

Temperature Calculation

Calculating the temperature at which a gas exerts a particular pressure in a fixed volume involves the use of the Ideal Gas Law, where temperature is a fundamental component. - **Ideal Gas Law Overview:** The equation \( PV = nRT \) connects a gas's pressure (P), volume (V), moles (n), and temperature (T), with R being the gas constant. - **Solving for Temperature:** To find the temperature, rearrange the formula to \( T = \frac{PV}{nR} \). All other variables are substituted into this equation to solve for T. In the given problem, after determining the number of moles and converting the pressure to atmospheres, this rearranged equation was used to find the temperature. The resulting temperature was calculated to be approximately 243.86 K. Understanding how each variable affects the others in this equation allows you to predict how changes in one condition could affect temperature.