Using Graham's law of effusion, if #"CO"_2# takes 32 sec to effuse, how long will hydrogen take?

Question

using Graham's law of effusion, if co2 takes 32sec to effuse how long will the hydrogen take?

Wyatt Adkins · Accepted Answer

#"7.7 s"#

According to Graham's Law of Effusion, a gas's rate of effusion is inversely related to the mass of its constituent particles squared.

This can be expressed as

#color(blue)("rate of effusion " prop color(white)(a)1/sqrt("molar mass"))#

In essence, a gas's effusion rate is determined by the mass of its molecules; the heavier the molecules, the slower the effusion rate.

Similarly, the rate of effusion increases with molecule weight.

Right from the start, you should look at your two gases, carbon dioxide, #"CO"_2#, and hydrogen gas, #"H"_2#, and be able to predict that it will take less time for the hydrogen gas to effuse when compared with the carbon dioxide.

Because hydrogen gas molecules are lighter and smaller than carbon dioxide molecules, this prediction is accurate.

Find the two gases' molar masses*.

#"For CO"_2: " 44.01 g mol"^(-1)#

#"For H"_2: " 2.016 g mol"^(-1)#

The two gases' rates of effusion can be expressed as

#"rate"_(CO_2) prop 1/sqrt("44.01 g mol"^(-1))#

additionally

#"rate"_(H_2) prop 1/sqrt("2.016 g mol"^(-1))#

This implies that you are capable of writing.

#"rate"_(CO_2)/"rate"_(H_2) = 1/sqrt("44.01 g mol"^(-1)) * sqrt("2.016 g mol"^(-1))/1#

#"rate"_(CO_2)/"rate"_(H_2) = sqrt((2.016 color(red)(cancel(color(black)("g mol"^(-1)))))/(44.01color(red)(cancel(color(black)("g mol"^(-1)))))) = 0.214#

This indicates that you've

#"rate"_(H_2) = "rate"_(CO_2) * 1/0.241#

#"rate"_(H_2) = 4.15 * "rate"_(CO_2)#

Hydrogen gas will release more quickly than carbon dioxide, as expected.

Therefore, if it takes #"32 s"# for carbon dioxide to effuse, and hydrogen effuses #4.15# times faster, you can say that you have

#t_(H_2) = "32 s"/4.15 = color(green)("7.7 s")#

Two sig figs are used to round the result.

SIDE NOTE: Depending on the values you choose for the molar masses of the two gases, the answer will change slightly.

If other values are given to you, for example #"44 g mol"^(-1)# and #"2.0 g mol"^(-1)#, simply redo the calculations and follow the same steps shown here.

Cooper Anderson · Answer

undefined Using Graham's law of effusion, if "CO"_2 takes 32 seconds to effuse, hydrogen, which has a lower molar mass than CO2, will take less time to effuse. The ratio of the effusion rates is the square root of the inverse ratio of their molar masses. Since the molar mass of CO2 is about 44 g/mol and the molar mass of hydrogen is about 2 g/mol, the ratio of their molar masses is 22. Therefore, the square root of the inverse ratio is √(1/22). Thus, hydrogen will take approximately 4.7 seconds to effuse.