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At a certain temperature the speeds of six gaseous molecules in a container are $2.0 \mathrm{m} / \mathrm{s}, 2.2 \mathrm{m} / \mathrm{s}, 2.6 \mathrm{m} / \mathrm{s}$ $2.7 \mathrm{m} / \mathrm{s}, 3.3 \mathrm{m} / \mathrm{s},$ and $3.5 \mathrm{m} / \mathrm{s} .$ Calculate the root-mean-square speed and the average speed of the molecules. These two average values are close to each other, but the root-mean-square value is always the larger of the two. Why?

The rms value is always larger than the avg because in rms the values are squared and added whereas in avg they are only added. Hence evenafter rooting the rms is larger than the avg.

Chemistry 101

Chapter 5

Gases

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Lectures

05:03

In physics, a gas is one of the three major states of matter (the others being liquid and solid). A gas is a fluid that does not support tensile stress, meaning that it is compressible. The word gas is a neologism first used by the early 17th-century Flemish chemist J.B. van Helmont, based on the Greek word ("chaos"), the simplest of all the elemental forms of matter.

04:46

In physics, thermodynamics is the science of energy and its transformations. The three laws of thermodynamics state that energy can be exchanged between physical systems as heat and work; that the total energy of a system can be calculated by adding up all forms of energy in the system; that energy spontaneously flows from being localized to becoming dispersed, spread out, or uniform; and that the entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.

03:06

At a certain temperature t…

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The root mean square veloc…

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The molecules in a six-par…

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How does the root-mean-squ…

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Two containers are at the …

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Why is the root-mean-squar…

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06:43

The average speed of molec…

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Determine the root-mean-sq…

01:38

I Eleven molecules have sp…

05:24

So to calculate the, uh, root mean square speed, we first square each value. And so now I have written the squares in, and, uh, now we take the mean of the squares. We add up all the squares and then we're going to divide by the total number of values, which is six. And once I do that, then I take the square root and I get to 0.77 in that. And that's all. The values were in leaders per second, so I'd square. I mean, get major squared per second squared. And then when I take the square root back two meters per second, the the average speed is just add up all of the values and divide by six. And indeed, the result is close to the root mean square speed on the question asks. Well, why is the root mean square speed always larger and essentially in taking the squares? Uh, before you divide by six when you take that average. In essence, each of the values has been weighted by itself and said the larger values, uh, 3.5 is multiplied by 3.5, but to is only multiplied by two And so it's as if where it is actually that the greater values are weighted more, they essentially count more, and so the overall average is going to be greater.

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