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If $\mathrm{v}_{\mathrm{rms}}=$ root mean square speed of molecules
$\mathrm{v}_{\mathrm{av}}=$ average speed of molecules
$\mathrm{v}_{\mathrm{mp}}=$ most probable speed of molecules Then, identify the correct relation between these speeds.
(A) $\mathrm{v}_{\mathrm{rms}}>\mathrm{v}_{\mathrm{ar}}>\mathrm{v}_{\mathrm{mp}}$
(B) $\mathrm{v}_{\mathrm{av}}>\mathrm{v}_{\mathrm{mp}}>\mathrm{v}_{\mathrm{rms}}$
(C) $\mathrm{v}_{\mathrm{mp}}>\mathrm{v}_{\mathrm{av}}>\mathrm{v}_{\mathrm{rms}}$
(D) $\mathrm{v}_{\mathrm{rms}}>\mathrm{v}_{\mathrm{av}}=\mathrm{v}_{\mathrm{mp}}$

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Last updated date: 17th Jun 2024
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Answer
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Hint: We know that the speed of the molecules in a gas is proportional to the temperature and is inversely proportional to molar mass of the gas. In other words, as the temperature of a sample of gas is increased, the molecules speed up and the root mean square molecular speed increases as a result. The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas. The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect the kinetic energy of a material.

Complete step by step answer
We know that A molecule is defined as two or more atoms of the same element that are bound together. A molecule may be homonuclear, which means, it consists of atoms of one chemical element, as with oxygen $\left(\mathrm{O}_{2}\right);$ or it may be heteronuclear, a chemical compound composed of more than one element, as with water $\left(\mathrm{H}_{2} \mathrm{O}\right)$.
The root mean square velocity is the square root of the average of the square of the velocity. The reason we use the rms velocity instead of the average is that for a typical gas sample the net velocity is zero since the particles are moving in all directions.
A molecule exists when two or more atoms join together by forming chemical bonds. When atoms of at least two different elements come together to form chemical bonds, these molecules can be called compounds. Sodium chloride (NaCl) is a classic example of an ionic compound, or compound formed by ionic bonds.
We know that root mean square speed of molecules > average speed of molecules > most probable speed of molecules.

So, the answer is option A.

Note: We can say that when heat is added to a substance, the molecules and atoms vibrate faster. As atoms vibrate faster, the space between atoms increases. The motion and spacing of the particles determine the state of matter of the substance. The end result of increased molecular motion is that the object expands and takes up more space. With an increase in temperature, the particles gain kinetic energy and move faster. The actual average speed of the particles depends on their mass as well as the temperature heavier particles move more slowly than lighter ones at the same temperature.