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Chapter 5: Problem 120

The rate of effusion of a particular gas was measured and found to be 24.0 mL/min. Under the same conditions, the rate of effusion of pure methane \(\left(\mathrm{CH}_{4}\right)\) gas is 47.8 mL/min. What is the molar mass of the unknown gas?

Short Answer

Expert verified
The molar mass of the unknown gas is approximately \(44.97 \: g/mol\), calculated using Graham's law of effusion formula and the given rates of effusion.

Step by step solution

01

Write down given values and Graham's law formula

To solve this problem, we should first identify the given values: Rate of effusion of unknown gas (R1) = 24.0 mL/min, Rate of effusion of methane gas (R2) = 47.8 mL/min, Molar mass of methane gas (M2) = 16 g/mol (since methane is \(\mathrm{CH}_{4}\) with 1 carbon and 4 hydrogen atoms, C = 12 g/mol and H = 1g/mol). Graham's law of effusion formula: \( \frac{R1}{R2} = \sqrt{\frac{M2}{M1}} \) Where R1 is the rate of effusion of the unknown gas, R2 is the rate of effusion of methane gas, M1 is the molar mass of the unknown gas, and M2 is the molar mass of methane gas.
02

Calculate the ratio of the two rates

According to Graham's law, we have to find the ratio between the rate of effusion of the unknown gas and methane gas: \( \frac{R1}{R2} = \frac{24.0}{47.8} \)
03

Solve for the molar mass of the unknown gas

We have the formula \( \frac{R1}{R2} = \sqrt{\frac{M2}{M1}} \) and calculated ratio \(\frac{R1}{R2}\) in step 2. Now, we will find the molar mass of the unknown gas (M1): \( \sqrt{\frac{M2}{M1}} = \frac{24.0}{47.8} \) Square both sides of the equation and solve for M1: \( \frac{M2}{M1} = \left(\frac{24.0}{47.8}\right)^2 \) \( M1 = \frac{M2}{(\frac{24.0}{47.8})^2} \) Use the given molar mass of methane gas (M2) = 16 g/mol to calculate M1: \( M1 = \frac{16}{(\frac{24.0}{47.8})^2} \)
04

Calculate the molar mass of the unknown gas

Now, we can calculate molar mass of the unknown gas (M1): \( M1 = \frac{16}{(\frac{24.0}{47.8})^2} = 44.97 \: g/mol \) The molar mass of the unknown gas is approximately 44.97 g/mol.

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Most popular questions from this chapter

Silane, SiH, , is the silicon analogue of methane, \(\mathrm{CH}_{4}\) . It is prepared industrially according to the following equations: $$\begin{array}{c}{\mathrm{Si}(s)+3 \mathrm{HCl}(g) \longrightarrow \mathrm{HSiCl}_{3}(l)+\mathrm{H}_{2}(g)} \\ {4 \mathrm{HSiCl}_{3}(l) \longrightarrow \mathrm{SiH}_{4}(g)+3 \mathrm{SiCl}_{4}(l)}\end{array}$$ a. If \(156 \mathrm{mL}\) \(\mathrm{HSiCl}_{3} (d=1.34 \mathrm{g} / \mathrm{mL})\) is isolated when 15.0 \(\mathrm{L}\) \(\mathrm{HCl}\) at 10.0 \(\mathrm{atm}\) and \(35^{\circ} \mathrm{C}\) is used, what is the percent yield of \(\mathrm{HSiCl}_{3} ?\) b. When \(156 \mathrm{HSiCl}_{3}\) is heated, what volume of \(\mathrm{SiH}_{4}\) at 10.0 \(\mathrm{atm}\) and \(35^{\circ} \mathrm{C}\) will be obtained if the percent yield of the reaction is 93.1\(\% ?\)

A \(1.0-\mathrm{L}\) sample of air is collected at \(25^{\circ} \mathrm{C}\) at sea level \((1.00 \mathrm{atm}) .\) Estimate the volume this sample of air would have at an altitude of 15 \(\mathrm{km}\) ( see Fig.5.30) .At \(15 \mathrm{km},\) the pressure is about 0.1 \(\mathrm{atm} .\)

Consider separate \(2.5-\) L gaseous samples of \(\mathrm{He}\), \(\mathrm{N}_{2},\) and \(\mathrm{F}_{2},\) all at \(\mathrm{STP}\) and all acting ideally. Rank the gases in order of increasing average kinetic energy and in order of increasing average velocity.

Do all the molecules in a 1 -mole sample of \(\mathrm{CH}_{4}(g)\) have the same kinetic energy at 273 \(\mathrm{K}\)? Do all molecules in a 1 -mole sample of \(\mathrm{N}_{2}(g)\) have the same velocity at 546 \(\mathrm{K}\) ? Explain.

The total mass that can be lifted by a balloon is given by the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon. Consider a hot-air balloon that approximates a sphere 5.00 m in diameter and contains air heated to \(65^{\circ} \mathrm{C}\) . The surrounding air temperature is \(21^{\circ} \mathrm{C} .\) The pressure in the balloon is equal to the atmospheric pressure, which is 745 torr. a. What total mass can the balloon lift? Assume that the average molar mass of air is 29.0 g/mol. (Hint: Heated air is less dense than cool air.) b. If the balloon is filled with enough helium at \(21^{\circ} \mathrm{C}\) and 745 torr to achieve the same volume as in part a, what total mass can the balloon lift? c. What mass could the hot-air balloon in part a lift if it were on the ground in Denver, Colorado, where a typical atmospheric pressure is 630. torr?
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