What Temperature Equals Hydrogen's RMS Velocity to Oxygen's at 300K?

[mit_hacker]

Homework Statement

(Q) Hydrogen molecules have a mass of 2u and oxygen molecules have a mass of 32u, where u is defined as an atomic mass unit (u = 1.660540 \times 10^{-27}\; {\rm kg}). Compare a gas of hydrogen molecules to a gas of oxygen molecules.

At what gas temperature T_rms would the root-mean-square (rms) speed of a hydrogen molecule be equal to that of an oxygen molecule in a gas at 300 K?
State your answer numerically, in kelvins, to the nearest integer.

Homework Equations

v = sqrt(3kT/m) where k is Boltzmann's constant.

The Attempt at a Solution

Using the above equation gives that Temperature is 300/16 = 18.75K. I know that this is a crazy answer but please don't laugh

What's wrong with my solution?

 

[HallsofIvy]

I'll promise not to laugh if you will tell me why 18.75 K is a "crazy answer"!

 

[mit_hacker]

Because...

Thats almost close to absolute 0 and when i fed the answer in, it said "not quite" which I rake to be a euphemism for "this is nonsense!"

 

[jeffsong]

the question requests the "the nearest integer" your result is correct, but I think the anwser should be 19

 

[rwisz]

Your solution is incorrect because you have used the wrong formula. The correct equation for root-mean-square velocity is v_rms = sqrt(3RT/M), where R is the gas constant and M is the molar mass of the gas. Using this formula, we can calculate the temperature at which the rms speed of a hydrogen molecule is equal to that of an oxygen molecule at 300 K.

First, we need to calculate the molar mass of hydrogen and oxygen molecules. Since 1 mole of any gas contains Avogadro's number of particles, the molar mass of a gas is equal to its molecular mass in grams. Therefore, the molar mass of hydrogen is 2 g/mol and the molar mass of oxygen is 32 g/mol.

Now, we can plug these values into the formula and set the rms velocities equal to each other:

sqrt(3RT_h2/2) = sqrt(3RT_o2/32)

Simplifying and rearranging gives:

sqrt(RT_h2) = sqrt(RT_o2/16)

Squaring both sides gives:

RT_h2 = RT_o2/16

Substituting in the values for R and the molar masses, we get:

(8.314 J/mol*K)(T_h2/2) = (8.314 J/mol*K)(300 K/16)

Solving for T_h2 gives:

T_h2 = 300/8 = 37.5 K

Therefore, the temperature at which the rms velocity of a hydrogen molecule is equal to that of an oxygen molecule at 300 K is 37.5 K. This is significantly higher than your initial answer of 18.75 K, which shows the importance of using the correct formula and units in scientific calculations.