Physics Topics are also essential for space exploration, allowing scientists to study phenomena such as gravitational waves and cosmic rays.
What is the Formula for the Mean Speed of Molecules in a Gas?
Molecular velocity is a vector quantity The number of gas molecules in any container is very large. So the velocity vectors are oriented randomly in all possible directions. As a result, the resultant velocity vector must be zero. Consequently, the mean velocity of the molecules is also zero. Clearly, this zero-value is useless as it gives no information about the order of magnitude of the molecular velocities.
Alternatively, we may take the magnitudes only of the molecular velocities to calculate the mean. Certainly it is non-zero and a useful quantity. We also know, the molecules move in straight lines between collisions. So the magnitude of molecular velocity is actually the molecular speed. The calculated mean velocity is essentially the mean speed of the molecules. However, mean speed is often loosely termed as mean velocity.
Let N be the number of molecules of a gas in a closed container and, at any instant,
c1, c2, c3, …… cN be the magnitudes of velocities of the N molecules, respectively.
So, mean velocity or mean speed of the molecules,
\(\bar{c}\) = \(\frac{c_1+c_2+c_3+\cdots+c_N}{N}\)
Mean square velocity of the molecules defined as the mean of the squares of velocities,
\(\overline{c^2}\) = \(\frac{c_1^2+c_2^2+c_3^2+\cdots+c_N^2}{N}\)
Root mean square speed or rms speed of the molecules, defined as the square-root of mean square speed,
C = \(\sqrt{\overline{c^2}}\) = \(\sqrt{\frac{c_1^2+c_2^2+c_3^2+\cdots+c_N^2}{N}}\)
The mean velocity E and the rms speed c of the gas molecules in a container are not equal. For example, let us take three molecules with velocities 40 m ᐧ s-1, 80 m ᐧ s-1 and 120 m ᐧ s-1. Then,
\(\bar{c}\) = \(\frac{40+80+120}{3}\) = 80 m ᐧ s-1
c = \(\sqrt{\frac{(40)^2+(80)^2+(120)^2}{3}}\) = 86.4 m m ᐧ s-1
In general, the rms speed is slightly greater than the mean velocity. In kinetic theory, the role of the rms speed is comparatively more important than that of the mean velocity.