Question

someone please help me, this is physics. :))))

Answer 1

show that

\[v_{avg} = \sqrt{\frac{8kT}{\pi m}}\]
\[v_{rms} = \sqrt{\frac{3kT}{m}}\]

Answer 2

\[v_{p} = \sqrt{\frac{2kT}{m}}\]

Answer 3

using the following,
\[\int\limits_{0}^{\infty} v^2 e^{-\lambda v^2} dv = \frac{1}{4} \sqrt{\frac{\pi}{\lambda^3}} \]
\[\int\limits_{0}^{\infty} v^3 e^{-\lambda v^2} dv = \frac{1}{2 \lambda^2}\]
\[\int\limits_{0}^{\infty} v^4 e^{-\lambda v^2} dv = \frac{3}{8} \sqrt{\frac{\pi}{\lambda^5}}\]

Answer 4

where \[v_{avg} = average speed\]
\[v_{rms} = root mean square speed\]
\[v_{p} = mostprobablespeed\]

Answer 5

Start with the Boltzmann distribution for the speed of particles in an ideal gas at fixed temperature T\[P(v) dv = N v^2 e^{-{mv^2} \over {2kT}} dv\]

Let \[\lambda = m/KT\]

Integrate from 0 to infinity and set the answer =1 to find N.
multiply by v and integrate to find the average speed.
multiply by v^2, integrate, and take a square root to find v_rms