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By what factor will the RMS speed of gas molecules increase if the temperature is increased from 50ºC to 170ºC?

Answer :

The rms speed of the gas molecules will increase by a factor of approximately 1.171 when the temperature is increased from 50ºC to 170ºC.

  • The rms speed, [tex]v_{rms}[/tex], of gas molecules is given by the formula:

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

  • where:

[tex]T[/tex] is the absolute temperature (in Kelvin),

[tex]k[/tex] is the Boltzmann constant,

[tex]m[/tex] is the mass of a gas molecule.

  • From this equation, you can see that:

[tex]v_{rms} \propto \sqrt{T}[/tex]

This means the rms speed is proportional to the square root of the absolute temperature.

  • First, convert the temperatures from Celsius to Kelvin:

Initial temperature, [tex]T_1[/tex]:

[tex]50^\circ C = 50 + 273 = 323 K[/tex]

Final temperature, [tex]T_2[/tex]:

[tex]170^\circ C = 170 + 273 = 443 K[/tex]

  • Next, calculate the factor of temperature increase in Kelvin:

[tex]\text{Factor} = \frac{T_2}{T_1} = \frac{443}{323} \approx 1.372[/tex]

  • Since [tex]v_{rms} \propto \sqrt{T}[/tex], we now find the factor by which the rms speed increases:

[tex]\text{Factor for } v_{rms} = \sqrt{\frac{443}{323}} = \sqrt{1.372} \approx 1.171[/tex]

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