Question

Which of the following graphs represents the velocity distribution of nitrogen molecules at 800K?

The graph below represents the velocity distribution of helium atoms at 800 K. The molar mass of helium gas is 0.004 kg/mole. The molar mass of nitrogen gas is 0.028 kg/mole. Which of the following graphs represents the velocity distribution of nitrogen molecules at 800 K?

Answer 1

There's an easy way to do it... `"N"_2` molecules are heavier than `"He"` atoms, so they will move more slowly. Hence, the most probable speed is slower , and as a result, the peak of the graph shifts to the left.

And since the speeds will in general be slower, to maintain the area under the curve, the speed distribution will also be more narrow.

Thus, we expect `"N"_2` to be a narrower distribution, and shifted to the left. Thus, we pick `C`.

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To prove it, I'll generate a graph of the Maxwell-Boltzmann distribution for `"N"_2` and we'll see what it matches. The distribution is given by:

`f(v) = 4pi (m/(2pik_BT))^(3//2)v^2e^(-mv^2//2k_BT)`

I'll plot the speed of `"He"` and of `"N"_2` from `"0 m/s"` to `"5000 m/s"`, at `"800 K"`, based on the mass of one particle of each.

`m_(He) = "4.0026 g"/"mol" xx "1 mol"/(6.0221413 xx 10^23 "particle") xx "1 kg"/"1000 g"`

`= 6.646 xx 10^(-27) "kg"`

`m_(N_2) = "28.014 g"/"mol" xx "1 mol"/(6.0221413 xx 10^23 "particle") xx "1 kg"/"1000 g"`

`= 4.652 xx 10^(-26) "kg"`

Here we have also that `k_B = 1.38065 xx 10^(-23) "kg"cdot"m"^2"/s"^2cdot"K"`. From this, the following graphs were generated in Excel.