Question

Question #633fc

Answer 1

`"1060 km h"^(-1)`

Explanation:

The root-mean-square speed of a molecule in a gas is calculated using the absolute temperature of the gas, the molar mass of the gas, and the universal gas constant.

`color(black)(|bar(ul(color(blue)(v_"rms" = sqrt((3RT)/M_M))))|) " "`, where

`v_"rms"` - the root-mean-square speed of the molecule
`R` - the universal gas constant, used as `"8.314 J mol"^(-1)"K"^(-1)`
`T` - the absolute temperature of the gas
`M_M` - the molar mass of the gas

You're dealing with krypton, a noble gas that exists as atoms in the gaseous state. The molar mass of krypton is equal to `"83.798 g mol"^(-1)`.

Now, something important to keep track of before doing the calculations. The root-mean-square speed is expressed is meters per second, `"m s"^(-1)`.

This means that you're going to have to manipulate some units under the square root to get `"m s"^(-1)` as the units for `v_"rms"`.

More specifically, you're going to have to use the fact that

`"1 J" = 1 "kg m"^2"s"^(-2) " " " "color(orange)("(*)")`

So, plug in your values into the equation - do not forget to convert the temperature from degrees Celsius to Kelvin!

`v_"rms" = sqrt((3 * "8.314 J" color(red)(cancel(color(black)("mol"^(-1)))) * color(red)(cancel(color(black)("K"^(-1)))) * (273.15 + 20.0)color(red)(cancel(color(black)("K"))))/("83.798 g" color(red)(cancel(color(black)("mol"^(-1))))))`

`v_"rms" = sqrt((3 * 8.314 * 293.15)/83.798) * sqrt("J g"^(-1))`

Now focus on the units first. Use the conversion factor

`"1 kg" = 10^3"g"`

to write

`"J"/color(red)(cancel(color(black)("g"))) * (10^3color(red)(cancel(color(black)("g"))))/"1 kg" = 10^3 "J kg"^(-1)`

Use conversion factor `color(orange)("(*)")` to write

`10^3"J kg"^(-1) = 10^3 color(red)(cancel(color(black)("kg"))) "m"^2"s"^(-2) * color(red)(cancel(color(black)("kg"^(-1)))) = 10^3"m"^2"s"^(-2)`

Plug this back to find `v_"rms"`

`v_"rms" = sqrt((3 * 8.314 * 293.15)/83.798 * 10^3"m"^2"s"^(-2)`

`v_"rms" = "295.4 m s"^(-1)`

To convert this to kilometers per hour, `"km h"^(-1)`, use the conversion factors

`"1 km" = 10^3"m" " "` and `" " "1 h " = " 3600 s"`

You will thus have

`295.4 color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"))) * "1 km"/(10^3color(red)(cancel(color(black)("m")))) * (3600color(red)(cancel(color(black)("s"))))/"1 h" = "1063.44 km h"^(-1)`

Rounded to three sig figs, the number of sig figs you have for the temperature of the gas, the answer will be

`v_"rms" = color(green)(|bar(ul(color(white)(a/a)"1060 km h"^(-1)color(white)(a/a)))|)`