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What is the average speed (actually the root-mean-square speed) of a carbon monoxide molecule at 27ºC?

A. 450 m/s
B. 412 m/s
C. 514 m/s
D. 517 m/s

Answer :

c. 514 m/s is the average speed of a carbon monoxide molecule at 27ºC.

The formula for rms speed:

[tex]U_{rms} = \sqrt{\frac{3kT}{m}}[/tex]

Where:

[tex]U_{rms}[/tex] is the root-mean-square speed.

[tex]k[/tex] is the Boltzmann constant, approximately [tex]1.38 \times 10^{-23} \, J/K[/tex].

[tex]T[/tex] is the temperature in Kelvin.

[tex]m[/tex] is the mass of a carbon monoxide molecule in kilograms.

Convert temperature from Celsius to Kelvin:
[tex]T = 27^{\circ}C + 273.15 = 300.15 \, K[/tex]

  • The mass of carbon monoxide (CO):

The molar mass of CO is approximately 28.01 g/mol. To convert to kg:
[tex]m = \frac{28.01 \, g/mol}{1000} = 0.02801 \, kg/mol[/tex]
Now, using the molar mass and Avogadro's number (approximately [tex]6.022 \times 10^{23} \, molecules/mol[/tex]), the mass of a single CO molecule is:
[tex]m = \frac{0.02801 \, kg/mol}{6.022 \times 10^{23} \, molecules/mol} \approx 4.65 \times 10^{-26} \, kg[/tex]

Substitute into the rms speed formula:
[tex]U_{rms} = \sqrt{\frac{3 \times (1.38 \times 10^{-23}) \times (300.15)}{4.65 \times 10^{-26}}} \approx 514 \, m/s[/tex].

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