How Does Molar Mass Affect Gas Molecule Speeds at Equal Temperatures?

[Gimp]

If the translational rms speed of the water vapor molecules (H2O) in air is 676 m/s, what is the translational rms speed of the carbon dioxide molecules (CO2) in the same air? Both gases are at the same temperature.

So what I have so far...

VrmsH20 = squareroot of 3RT/mCO2
VrmsCO2 = squareroot of 3RT/mCO2

676 = squareroot of mCO2/mH2O

And I already found the mass of the CO2 and H2O, but then I'm stuck... Can anyone help please?

 

[Psi-String]

$$V_{rms} = \sqrt{\frac{3RT}{M}}$$

since the temperature is the same, think proportion.

 

[quicknote]

Based on the given information, we can use the root mean square (rms) formula for gases to find the translational rms speed of CO2 molecules in the same air. The formula is Vrms = √(3RT/m), where Vrms is the translational rms speed, R is the gas constant (8.314 J/mol*K), T is the temperature (in Kelvin), and m is the molar mass of the gas.

Since both gases are at the same temperature, we can use the same value for T in the formula. However, we need to find the molar mass of CO2 in order to calculate its translational rms speed.

The molar mass of CO2 is 44.01 g/mol, while the molar mass of H2O is 18.02 g/mol. Plugging these values into the formula, we get:

VrmsCO2 = √(3*8.314*298/44.01) = 401.2 m/s

Therefore, the translational rms speed of CO2 molecules in the same air is approximately 401.2 m/s. This is significantly lower than the translational rms speed of H2O molecules (676 m/s) due to the difference in molar mass between the two gases.