R.m.s. velocity of a gas molecule of mass m at given temperature is proportional to _______________

The root mean square (RMS) velocity of a gas molecule at a given temperature is directly proportional to the square root of the temperature. To understand this relationship better, let's break it down step by step.

The Relationship Between RMS Velocity and Temperature

In the kinetic theory of gases, the RMS velocity (v_rms) is a measure of the average speed of the molecules in a gas. It can be expressed mathematically as:

v_rms = √(3kT/m)

In this equation:

• v_rms is the root mean square velocity.
• k is the Boltzmann constant.
• T is the absolute temperature measured in Kelvin.
• m is the mass of a gas molecule.

Understanding the Proportionality

From the equation, you can see that the RMS velocity is proportional to the square root of the temperature (T). This means that if the temperature of the gas increases, the RMS velocity increases as well. Specifically:

• If you double the temperature, the RMS velocity increases by a factor of √2.
• Conversely, if the temperature decreases, the RMS velocity decreases accordingly.

Why Temperature Matters

Temperature is a measure of the average kinetic energy of the gas molecules. When you heat a gas, you are essentially giving its molecules more energy, which results in them moving faster. The kinetic energy of a molecule can be expressed as:

E_k = (1/2)mv^2

From this, we can derive that as kinetic energy increases (due to an increase in temperature), the velocity of the molecules must also increase, which is reflected in the RMS velocity formula.

Real-World Analogy

Think of a group of people at a party. If the music gets louder and the atmosphere becomes more energetic (similar to an increase in temperature), people will start dancing faster and moving around more energetically. This is analogous to gas molecules speeding up as temperature rises.

In summary, to answer your question, the RMS velocity of a gas molecule of mass m at a given temperature is proportional to the square root of the temperature. This relationship highlights how temperature influences the kinetic energy and movement of gas molecules, which is fundamental in understanding gas behavior in various scientific contexts.