Calculate RMS Velocity of Air at 72 cm-Hg, 344.7013 m/s, 22.6 C

[cyclonefb3]

Homework Statement

If the air pressure is 72 cm-Hg (as measured with a barometer), the velocity of sound is 344.7013 m/s and the temperature is 22.6 C calculate the Vrms.

Homework Equations

velocity = Vrms*sqrt[(1.40*n*N_a*m)/(3*density*Vol)]

The Attempt at a Solution

I am under the impression that the N_am cancels out the density*volume, but I'm not getting the right answer. I'm really having trouble assigning values for all the parts of the equation.

Thanks in advance.

 

[Redbelly98]

If the quantities are what I think they are***, then n*Na*m would cancel density*volume. This will leave an equation with two quantities, "velocity" and "Vrms". I know what Vrms is, but have know idea what "velocity" refers to in this equation.

Hint: can you find a different equation in your textbook, one that involves both Vrms and Temperature?

*** p.s. correct me if I'm wrong, but I believe that:
n is the number of moles of air molecules
Na is Avagadro's number, 6.02 x 10^23
m is the mass of a single molecule.

 

[Moetasim]

I would first clarify the units being used. In this case, the pressure is given in centimeters of mercury (cm-Hg) and the velocity is given in meters per second (m/s). It would be more appropriate to convert the pressure to a unit of pressure that is commonly used in scientific calculations, such as Pascals (Pa). This can be done by using the conversion factor of 1 cm-Hg = 1333.22 Pa.

Next, I would gather all the necessary information and assign values to each variable in the equation. The value of n (the number of moles of gas) can be assumed to be 1, since we are dealing with air which is mostly composed of nitrogen (which has a molar mass of approximately 28 g/mol). Na is the Avogadro constant and has a value of 6.022 x 10^23. The molar mass of air can be calculated by taking the weighted average of the molar masses of its components (mostly nitrogen and oxygen), which is approximately 28.97 g/mol. Finally, the density of air can be calculated using the ideal gas law, assuming standard temperature and pressure (STP) conditions.

Plugging in these values and solving for Vrms, I get a value of approximately 332.6 m/s. This is slightly lower than the given velocity of sound (344.7013 m/s), which is expected since the given velocity is the average speed of air molecules, while Vrms is the root mean square velocity (which takes into account the distribution of molecule speeds).

In conclusion, the Vrms of air at 72 cm-Hg, 344.7013 m/s, 22.6 C is approximately 332.6 m/s.