An incandescent light bulb is filled with neon gas. The gas that is in close proximity to the element of the bulb is at a temperature of 2300 Kelvin. Determine the root-mean-square speed of neon atoms in close proximity to the element. Use a value of 20.2 grams per mole for the molar mass of neon.
In this scenario, we have a bulb filled with neon gas. We’ll say it’s a pink neon. And we’re told that for atoms very close to the heating element, they have a temperature we can call capital 𝑇 of 2300 Kelvin. To solve for the speed, called a root-mean-square speed, we can use the relationship that the speed of a gas is equal to the square root of three times the gas constant 𝑅 multiplied by the temperature of the gas 𝑇, in Kelvin, all divided by the molar mass of that gas.
In the problem statement, we’re told the molar mass of neon. And we can look up the value of the gas constant 8.314 joules every mole Kelvin. Given the fact we also know the temperature 𝑇 of the neon atoms close to the element, we have all the information we need to plug in to this equation for root-mean-square speed. Before we do, we’ll be better able to understand our result by first zooming in on the neon gas inside this light bulb. Were we to do this, we would see all sorts of neon atoms bouncing around inside our light bulb at all sorts of speeds in all directions.
If we were to write down the velocities of each one of these particles, we know that these velocities will be both positive and negative since they point in all directions. To avoid negative values, we can square the results we get for the velocity of each atom. And next, we can average all of these squared velocities together. The point here is that some of these velocities are very great, some of them are very small. But we want a representative sample. So we take an average. Then finally, since we’ve squared this value, to counteract that operation, we take the square root. And we end up with what’s called the root-mean-square speed.
One way we can tell it’s a speed and not a velocity is that this result will always be positive. So that’s what we’re after, we’re after the root-mean-square speed of the neon atoms inside this light bulb. And now that we know more about what we’re calculating, we can go ahead and enter the numbers in.
When we do enter in our values for the gas constant 𝑅, our temperature in Kelvin, and our molar mass, we’re careful to convert the molar mass into units of kilograms so it’s consistent with the rest of our units. Calculating this value, we find, to two significant figures, it’s 1.7 times 10 to the third meters per second. That’s the square root of the average of the squares of the speeds of all the individual neon atoms.