Question #633fc
1 Answer
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
You're dealing with krypton, a noble gas that exists as atoms in the gaseous state. The molar mass of krypton is equal to
Now, something important to keep track of before doing the calculations. The root-mean-square speed is expressed is meters per second,
This means that you're going to have to manipulate some units under the square root to get
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
#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" = 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,
#"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)))|)#