The molecular velocity of any gas is
(a) inversely proportional to the square root of temperature
(b

Q: Why are there three different types of velocities?

Because molecules in a gas move at a wide range of different speeds, we need statistical averages to describe the macroscopic properties of the gas. $v_{rms}$ is used for energy calculations, $v_{avg}$ is used for collision rates, and $v_{mp}$ represents the most likely speed a random molecule will have.

Q: Does this proportionality apply to all gases?

Yes, this relationship is derived for ideal gases. Real gases behave very much like ideal gases at high temperatures and low pressures, so this proportionality holds true for them under those conditions.

Q: What happens to the velocity if we double the temperature?

Since velocity is proportional to the square root of temperature ($v \propto \sqrt{T}$), doubling the temperature ($2T$) will increase the velocity by a factor of $\sqrt{2}$, which is approximately 1.414 times the original velocity.

Q: Why is it proportional to the square root and not directly to $T$?

Temperature is a measure of average kinetic energy ($E_k = \frac{3}{2}kT$). Since kinetic energy is proportional to velocity squared ($E_k = \frac{1}{2}mv^2$), we get $v^2 \propto T$. Taking the square root of both sides gives $v \propto \sqrt{T}$.

Q: Does the mass of the gas affect this proportionality?

The molar mass $M$ affects the absolute value of the velocity (lighter gases move faster because $v \propto 1/\sqrt{M}$). However, for any given gas, the mass is constant, so the proportionality with temperature ($v \propto \sqrt{T}$) remains exactly the same.

Question

The molecular velocity of any gas is

Options:

(A) inversely proportional to the square root of temperature

(B) inversely proportional to absolute temperature

(C) directly proportional to square of temperature

(D) directly proportional to square root of temperature

Sigma Percentile · Accepted Answer

### The Kinetic Theory of Gases

Imagine a closed container filled with a gas. Inside, billions of tiny molecules are zipping around in constant, random motion. They collide with each other and with the walls of the container. This chaotic dance is the foundation of the **Kinetic Theory of Gases**.

One of the most profound insights of this theory is the physical meaning of temperature. Temperature is not just a number on a thermometer; it is a direct measure of the **average kinetic energy** of these gas molecules. The hotter the gas, the more energetic and faster the molecules become.

### The Maxwell-Boltzmann Distribution

Even though the gas has a specific temperature, not all molecules move at the exact same speed. Some are sluggish, some are incredibly fast, but most travel at an intermediate speed. This spread of speeds is beautifully described by the **Maxwell-Boltzmann Distribution**.

If you look at the graph of this distribution, the x-axis represents the molecular velocity ($v$), and the y-axis represents the fraction of molecules having that velocity ($f(v)$). At a given temperature $T_1$, the curve has a distinct peak. This peak corresponds to the **most probable velocity** ($v_{mp}$)—the speed possessed by the largest number of molecules.

When we heat the gas to a higher temperature $T_2$, we inject more thermal energy into the system. The entire curve shifts to the right and flattens out. This visual shift perfectly illustrates that at higher temperatures, the molecules are, on average, moving much faster.

### Types of Molecular Velocities

Because of this wide distribution of speeds, physicists define three different types of statistical velocities to describe the gas:

1. **Root Mean Square Velocity ($v_{rms}$):** This is the square root of the average of the squares of the velocities. It is the most important velocity for calculating the kinetic energy of the gas.
   $$v_{rms} = \sqrt{\frac{3RT}{M}}$$

2. **Average Velocity ($v_{avg}$):** This is the simple arithmetic mean of the speeds of all molecules.
   $$v_{avg} = \sqrt{\frac{8RT}{\pi M}}$$

3. **Most Probable Velocity ($v_{mp}$):** As seen on the graph, this is the speed at the peak of the Maxwell-Boltzmann curve.
   $$v_{mp} = \sqrt{\frac{2RT}{M}}$$

In all these formulas, $R$ is the universal gas constant, $T$ is the absolute temperature in Kelvin, and $M$ is the molar mass of the gas.

### The Final Verdict

If you look closely at the mathematical structure of all three formulas, a glaring pattern emerges. Whether you are calculating $v_{rms}$, $v_{avg}$, or $v_{mp}$, the velocity $v$ is always proportional to the square root of the absolute temperature $T$ divided by the molar mass $M$.

$$v \propto \sqrt{\frac{T}{M}}$$

For any specific gas, its molar mass $M$ is constant. Therefore, the relationship between its molecular velocity and temperature simplifies to:

**$$v \propto \sqrt{T}$$**

This means that the molecular velocity of any gas is **directly proportional to the square root of its absolute temperature**. If you want to double the speed of the molecules, you don't just double the temperature; you have to quadruple it! This elegant square-root relationship is a fundamental pillar of thermodynamics and a highly scoring concept in competitive exams.

Animated Solution for Chemistry - States of Matter: The molecular velocity of any gas is

Select Answer:

Visualized Solution

$f (v)$ vs $v$

$v_{r m s}, v_{a v g}, v_{m p}$

$v \propto T$

$T_{2} > T_{1}$

$Conclusion$

The Sigma Insight: Gaseous State

The Kinetic Theory of Gases

The Maxwell-Boltzmann Distribution

Types of Molecular Velocities

The Final Verdict

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