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Chapter 5: Problem 103

Based on your knowledge of the kinetic theory of gases, derive Graham's law of diffusion [Equation ( 5.17 )]

Short Answer

Expert verified
Given the average speed of a gas is \( v_{avg} = \sqrt{\frac{8kT}{\pi M}} \), the ratio of the rates of diffusion of two gases A and B is given by Graham's law \( \frac{r_A}{r_B} = \frac{v_{avg,A}}{v_{avg,B}} = \sqrt{\frac{M_B}{M_A}} \).

Step by step solution

01

Understanding the Kinetic Theory of Gases

The kinetic theory of gases assumes that gases consist of a large number of particles, all of which are in constant, random motion. The theory holds that the speed of these particles depends on their temperature, with higher temperatures resulting in faster speeds. The faster the particles move, the more frequently and forcefully they collide with each other, leading to a higher pressure. This relates to the rate of diffusion, since the faster particles move, the more quickly a gas will spread out and diffuse.
02

Defining Rate of Diffusion

The rate of diffusion or effusion of a gas is defined as the volume of the gas which passes through a small hole per unit time. It is given by the formula \( rate = \frac{Volume}{Time} \). This rate of diffusion is proportional to the average speed of the gas particles.
03

Relating Speed to Molar Mass

The average speed of gas particles can be given by the formula \( v_{avg} = \sqrt{\frac{8kT}{\pi M}} \) where \( v_{avg} \) is the average speed, \( k \) is Boltzmann constant and \( T \) is temperature in Kelvin, \( M \) is the molar mass of the gas in kg.
04

Derive Graham's Law

Consider two gases A and B with molar masses \( M_A \) and \( M_B \) respectively. The rates of diffusion \( r_A \) and \( r_B \) are given by the average speeds of the gas particles, so \( r_A = v_{avg,A} \) and \( r_B = v_{avg,B} \). Therefore, as \( r_A = \sqrt{\frac{8kT}{\pi M_A}} \) and \( r_B = \sqrt{\frac{8kT}{\pi M_B}} \), the ratio \( \frac{r_A}{r_B} = \frac{v_{avg,A}}{v_{avg,B}} = \sqrt{\frac{M_B}{M_A}} \) .This equation is Graham's law of diffusion.

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Most popular questions from this chapter

Some commercial drain cleaners contain two components: sodium hydroxide and aluminum powder. When the mixture is poured down a clogged drain, the following reaction occurs: $$2 \mathrm{NaOH}(a q)+2 \mathrm{Al}(s)+6 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow_{2 \mathrm{NaAl}(\mathrm{OH})_{4}(a q)+3 \mathrm{H}_{2}(g)}$$ The heat generated in this reaction helps melt away obstructions such as grease, and the hydrogen gas released stirs up the solids clogging the drain. Calculate the volume of \(\mathrm{H}_{2}\) formed at STP if \(3.12 \mathrm{~g}\) of \(\mathrm{Al}\) is treated with an excess of \(\mathrm{NaOH}\).

Explain why a helium weather balloon expands as it rises in the air. Assume that the temperature remains constant.

An ideal gas originally at 0.85 atm and \(66^{\circ} \mathrm{C}\) was allowed to expand until its final volume, pressure, and temperature were \(94 \mathrm{~mL}, 0.60 \mathrm{~atm},\) and \(45^{\circ} \mathrm{C},\) respectively. What was its initial volume?

Give two pieces of evidence to show that gases do not behave ideally under all conditions.

List the characteristics of an ideal gas.
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