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Chapter 6: Problem 127

At \(286 \mathrm{K},\) three gases, \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C},\) have root-mean-square speeds of \(360 \mathrm{m} / \mathrm{s}, 441 \mathrm{m} / \mathrm{s},\) and \(472 \mathrm{m} / \mathrm{s},\) respectively. Which gas is \(\mathrm{O}_{2} ?\)

Short Answer

Expert verified
Answer: Gas A is most likely to be O鈧�.

Step by step solution

01

Identify the formula for root-mean-square speed

We will use the formula for the root-mean-square speed, which is defined as: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) where \(v_{rms}\) is the root-mean-square speed, \(R\) is the ideal gas constant (8.314 J/(mol路K)), \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas in kg/mol.
02

Rearrange the formula to find the molar mass

To find the molar mass of each gas, we need to rearrange the formula: \(M = \frac{3RT}{v_{rms}^2}\)
03

Calculate the molar mass of each gas

Now, we can plug in the values for Gas A, Gas B, and Gas C to find their molar masses: Gas A: \(M_A = \frac{3(8.314 \, \text{J/(mol路K})(286 \, \text{K})}{(360 \, \text{m/s})^2} \approx 0.039 \, \text{kg/mol}\) Gas B: \(M_B = \frac{3(8.314 \, \text{J/(mol路K})(286 \, \text{K})}{(441 \, \text{m/s})^2} \approx 0.024 \, \text{kg/mol}\) Gas C: \(M_C = \frac{3(8.314 \, \text{J/(mol路K})(286 \, \text{K})}{(472 \, \text{m/s})^2} \approx 0.022 \, \text{kg/mol}\)
04

Determine which gas is O鈧�

The molar mass of \(\mathrm{O}_{2}\) is about 0.032 kg/mol (from the periodic table). By comparing the calculated molar masses of the gases, we can determine that Gas A (with a molar mass of approximately 0.039 kg/mol) is the closest to the molar mass of \(\mathrm{O}_{2}\). Therefore, the gas A is most likely to be \(\mathrm{O}_{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Constant
The ideal gas constant, symbolized as R, is a cornerstone in the study of thermodynamics and chemistry. It appears in the fundamental equation for the root-mean-square speed of gases, and is equally important in the ideal gas law, which is written as PV=nRT, where P is pressure, V volume, n the number of moles, and T temperature.

In terms of units, R has the value of 8.314 J/(mol路K), which means each mole of an ideal gas has an energy of 8.314 Joules per Kelvin per mole. Understanding this constant is key to solving a variety of problems involving gases, including calculating the root-mean-square speed, as it links together temperature, energy, and amount of gas.
Molar Mass Calculation
Molar mass is a measure of the mass of one mole of a substance, which is defined as exactly 6.022脳10虏鲁 particles of that substance (Avogadro's number). The molar mass is typically expressed in grams per mole (g/mol) or kilograms per mole (kg/mol), depending on the context.

To calculate the molar mass of a gas when given its root-mean-square speed, temperature, and the ideal gas constant, we can rearrange the formula for root-mean-square speed, which gives us a way to link kinetic theory with measurable quantities. By isolating molar mass on one side of the equation鈥攁fter plugging in the values for the ideal gas constant, temperature, and root-mean-square speed鈥攚e can solve for this vital piece of information that reflects the average weight of a mole's worth of molecules.
Gaseous Molar Masses
Understanding gaseous molar masses is crucial when working with gases and developing an insight into their behavior under different conditions. Every gas has a characteristic molar mass that can be found using the periodic table, or in the case of compounds, by summing the molar masses of its constituent elements.

For example, in the context of our problem, having determined the molar mass of Gas A as close to the known molar mass of O鈧� (32 g/mol or 0.032 kg/mol), we make an informed guess about its identity. Knowing the molar masses of various gases also allows us to predict their behavior under changing conditions using the ideal gas law, deduce rates of diffusion and effusion through Graham's law, and solve for unknown variables in chemical equations.

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Most popular questions from this chapter

The rate of effusion of an unknown gas is \(0.10 \mathrm{m} / \mathrm{s}\) and the rate of effusion of \(\mathrm{SO}_{3}(g)\) is \(0.052 \mathrm{m} / \mathrm{s}\) under identical experimental conditions. What is the molar mass of the unknown gas?

The following reactions were carried out in sealed containers. Will the total pressure after each reaction is complete be greater than, less than, or equal to the total pressure before the reaction? Assume all reactants and products are gases at the same temperature. a. \(\mathrm{N}_{2} \mathrm{O}_{5}(g)+\mathrm{NO}_{2}(g) \rightarrow 3 \mathrm{NO}(g)+2 \mathrm{O}_{2}(g)\) b. \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{SO}_{3}(g)\) c. \(\mathrm{C}_{3} \mathrm{H}_{8}(g)+5 \mathrm{O}_{2}(g) \rightarrow 3 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)\) d. \(4 \mathrm{NH}_{3}(g)+5 \mathrm{O}_{2}(g) \rightarrow 4 \mathrm{NO}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)\)

One balloon was filled with \(\mathrm{H}_{2},\) another with He. The person responsible for filling them neglected to label them. After 24 h the volumes of both balloons had decreased but by different amounts. Which balloon contained hydrogen?

Which gas sample has the largest volume at \(25^{\circ} \mathrm{C}\) and 1 atm pressure? \((\mathrm{a}) 0.500\) mol of \(\mathrm{dry} \mathrm{H}_{2} ;\) (b) 0.500 mol of dry \(\mathrm{N}_{2} ;\) (c) 0.500 mol of wet \(\mathrm{H}_{2}\left(\mathrm{H}_{2} \text { collected over water }\right)\)

Students at the University of North Texas and the University of Washington built a car propelled by compressed nitrogen gas. The gas was obtained by boiling liquid nitrogen stored in a 182 - L tank. What volume of \(\mathrm{N}_{2}\) is released at 0.927 atm of pressure and \(25^{\circ} \mathrm{C}\) from a tank full of liquid \(\mathrm{N}_{2}(d=0.808 \mathrm{g} / \mathrm{mL}) ?\)
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