Question

Derive the Expression for Molecular velocities.​

Answer 1

Answer: Root mean square velocity vmn=(V2−)=3KBTm−−−−−−√, where 'm' is the mass of the molecule, k - Boltzmann constant and T is the absolute scale of temperature.

Answer 2

Answer:

Expression for the average molecular velocity of gas molecules are $v=\sqrt{8RT/\pi M}$  (where R is the gas constant, T is temperature and, M is the mass of the molecule)

Explanation:

• Formula used:

                                                  $\int\limits^ \infty_0 {vP(v)} \, dv$

• In this case, the probability density function P(v) is the Maxwell-Boltzmann distribution and the quantity we want to find the expectation value is simply the velocity v.

• . We know velocity can only be positive. So we have the limit from zero to ∞

• Therefore, the mean velocity is given by,

                                                    $\int\limits^ \infty_0 {vP(v)} \, dv$

• Where, P(v)is the Maxwell-Boltzmann distribution given as,

                                    $P(v) = (m/2\pi kT)^{3/2} 4\pi v^{2} e^{(-mv^{2}/2kT )}$

• Where m is the mass of the gas molecule. k is the Boltzmann constant, T is the given temperature, and v is the velocity of one molecule.

• So, the average velocity is given as,

                            $v = 4\pi (m/2\pi kT)^{3/2} \int\limits^\infty_0{v^{3} } e^{-mv^{2}/2kT} \, dv$

• The result of integral in this form is given as,

                                    $\int\limits^\infty_0 {v^{3} } exp(-\alpha v^{2} )dv = 1/2\alpha^{2}$

• So, here we have      $\alpha = m/2kT$

                             ⇒ $v=4\pi (m/2\pi kT)^{3/2} 4k^{2} T^{2} /2m^{2}$

                             ⇒$v=\sqrt[]{8kT/\pi m}$

• Here, m is the mass of one molecule. If we take the mass of the compound as M, they are related by M=Nm (Where N is the Avogadro's number)

• Since, R=Nk, where R is the gas constant, we can modify the equation as,

                                     $v=\sqrt{8RT/\pi M}$  

• So, we have found the average velocity of a gas molecule as

                                      $v=\sqrt{8RT/\pi M}$

Learn more about Maxwell-Boltzmann distribution:

https://brainly.in/question/1822855

Learn more about Avogadro's number:

https://brainly.in/question/762547

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