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Chapter 10: Problem 55

Magnesium can be used as a "getter" in evacuated enclosures to react with the last traces of oxygen. (The magnesium is usually heated by passing an electric current through a wire or ribbon of the metal.) If an enclosure of 0.452 L. has a partial pressure of \(\mathrm{O}_{2}\) of \(3.5 \times 10^{-6}\) torr at \(27^{\circ} \mathrm{C},\) what mass of magnesium will react according to the following equation? $$2 \mathrm{Mg}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{MgO}(s)$$

Short Answer

Expert verified
The mass of magnesium required to react with the oxygen in the enclosure is approximately \(3.38 \times 10^{-9}\) g.

Step by step solution

01

Convert given values to appropriate units

First, we need to convert the given values to appropriate units for calculation. The temperature of the system should be in Kelvin, and the pressure should be in atm (if we use R in atm.L/(mol.K)). - Convert temperature to Kelvin: T = 27°C + 273.15 = 300.15 K - Convert pressure to atm: P = \(3.5 \times 10^{-6}\) torr × (1 atm / 760 torr) ≈ \(4.61 \times 10^{-9}\) atm
02

Apply the Ideal Gas Law

Using the Ideal Gas Law (PV = nRT), we can calculate the amount of oxygen in moles. - V = 0.452 L (given) - R = 0.0821 atm.L/(mol.K) Solve for n: n = PV/(RT) = (\(4.61 \times 10^{-9}\) atm × 0.452 L) / (0.0821 atm.L/(mol.K) × 300.15 K)
03

Calculate the amount of oxygen in moles

Now let's calculate the amount of oxygen in moles using the numbers we obtained: n = (\(4.61 \times 10^{-9}\) atm × 0.452 L) / (0.0821 atm.L/(mol.K) × 300.15 K) ≈ \(6.95 \times 10^{-11}\) mol
04

Find the amount of magnesium required using stoichiometry

From the balanced chemical equation, we can see that 2 moles of magnesium react with 1 mole of oxygen: 2 Mg + O₂ → 2 MgO. Using stoichiometry, we can calculate the amount of magnesium required in moles: Amount of Mg in moles = 2 × Amount of O₂ in moles = 2 × \(6.95 \times 10^{-11}\) mol ≈ \(1.39 \times 10^{-10}\) mol
05

Calculate the mass of magnesium required

Now we can determine the mass of magnesium required by multiplying the number of moles by the molar mass of magnesium (24.31 g/mol): Mass of Mg = Amount of Mg in moles × Molar mass of Mg = \(1.39 \times 10^{-10}\) mol × 24.31 g/mol ≈ \(3.38 \times 10^{-9}\) g Therefore, the mass of magnesium required to react with the oxygen in the enclosure is approximately \(3.38 \times 10^{-9}\) g.

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Most popular questions from this chapter

(a) List two experimental conditions under which gases deviate from ideal behavior. (b) List two reasons why the gases deviate from ideal behavior.

Consider the following gases, all at STP: Ne, SF \(_{6}, \mathrm{N}_{2}, \mathrm{CH}_{4}\) . (a) Which gas is most likely to depart from the assumption of the kinetic-molecular theory that says there are no attractive or repulsive forces between molecules? (b) Which one is closest to an ideal gas in its behavior? (c) Which one has the highest root-mean-square molecular speed at a given temperature? (d) Which one has the highest total molecular volume relative to the space occupied by the gas? (e) Which has the highest average kinetic-molecular energy? (f) Which one would effuse more rapidly than \(\mathrm{N}_{2} ?\) (g) Which one would have the largest van der Waals \(b\) parameter?

Many gases are shipped in high-pressure containers. Consider a steel tank whose volume is 55.0 gallons that contains \(\mathrm{O}_{2}\) gas at a pressure of \(16,500 \mathrm{kPa}\) at \(23^{\circ} \mathrm{C}\) . (a) What mass of \(\mathrm{O}_{2}\) does the tank contain? (b) What volume would the gas occupy at STP? (c) At what temperature would the pressure in the tank equal 150.0 atm? (d) What would be the pressure of the gas, in kPa, if it were transferred to a container at \(24^{\circ} \mathrm{C}\) whose volume is 55.0 \(\mathrm{L}\) ?

In the Dumas-bulb technique for determining the molar mass of an unknown liquid, you vaporize the sample of a liquid that boils below \(100^{\circ} \mathrm{C}\) in a boiling-water bath and determine the mass of vapor required to fill the bulb. From the following data, calculate the molar mass of the unknown liquid: mass of unknown vapor, 1.012 g; volume of bulb, \(354 \mathrm{cm}^{3} ;\) pressure, 742 torr; temperature, \(99^{\circ} \mathrm{C}\) .

At constant pressure, the mean free path \((\lambda)\) of a gas molecule is directly proportional to temperature. At constant temperature, \(\lambda\) is inversely proportional to pressure. If you compare two different gas molecules at the same temperature and pressure, \(\lambda\) is inversely proportional to the square of the diameter of the gas molecules. Put these facts together to create a formula for the mean free path of a gas molecule with a proportionality constant (call it \(R_{\text { mfp }}\) , like the ideal-gas constant) and define units for \(R_{\operatorname{mfp}}\) .
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