Mechanical Engineering - Thermodynamics - Discussion

The most probable speed, vp, is the speed most likely to be possessed by any molecule (of the same mass m) in the system and corresponds to the maximum value or mode of f(v). To find it, we calculate df/dv, set it to zero and solve for v:

\frac{df(v)}{dv} = 0

which yields:

v_p = \sqrt {\frac{2kT}{m}} = \sqrt {\frac{2RT}{M}}


The root mean square speed is the second-order moment of speed:

= \sqrt {\langle v^2 \rangle} = \left(\int_0^{\infty} v^2 \, f(v) \, dv \right)^{1/2}.

= \sqrt {\frac{3kT}{m}} = \sqrt {\frac{3RT}{M} } = \sqrt{\frac{3}{2}} v_p.

The most probable speed, vp, is the speed most likely to be possessed by any molecule (of the same mass m) in the system and corresponds to the maximum value or mode of f(v). To find it, we calculate df/dv, set it to zero and solve for v:

\frac{df(v)}{dv} = 0

which yields:

v_p = \sqrt { \frac{2kT}{m} } = \sqrt { \frac{2RT}{M} }

Ref : http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution#Distribution_for_relative_speed

μ rms = √ (3RT/M).

where
μrms = root mean square velocity in m/sec.
R = ideal gas constant = 8.3145 (kg.m2/sec2)/Kmol.
T = absolute temperature in Kelvin.
M = mass of a mole of the gas in kilograms.

While here many got doubts about B & C options. I goggled it and found that the word 'Probable velocity makes the difference here.

Option : B is correct.

Option C is RMS velocity of molecules.

Answer B. Probable velocity of the gas molecules.
Answer C. Typical velocity of molecules in a gas.

In the question, they asked probable velocity so Answer B is right.

K.E is directly proportional to Temperature.

1/2mv^2 = kT.
v^2 = 2kT/m.
So v=√(2kT/m).
Where k= proportionality constant.

Then how to find most probable velocity?

Please mention the method just like you explained about root mean square velocity.

Answer 2 is correct. The third answer (C) is for root mean square velocity. And 2nd (B) is the most probable velocity.

Formula for most probable velocity.

vp = option B.

And for root mean square velocity.

Vrms=option C.

I guess it is option C. The rms velocity itself is most probable velocity. Please if any one clarify.