Various Speeds of Gas Molecules: Comprehensive NEET Physics Notes

1. Various Speeds of Gas Molecules

1.1 Introduction to Molecular Speeds

In kinetic theory, gas molecules are in constant, random motion, and their speeds vary based on factors like temperature and molecular mass. The behavior of these molecules can be described using three key types of speeds:

1. Root Mean Square (rms) Speed
2. Average Speed
3. Most Probable Speed

Each of these speeds provides a different statistical perspective on the motion of gas molecules.

1.2 Root Mean Square (rms) Speed

The root mean square speed is the square root of the mean of the squares of the speeds of all gas molecules. It is a direct indicator of the average kinetic energy of the gas molecules. The equation for rms speed is:

$v_{\text{rms}}=\sqrt{\frac{3k_{B}T}{m}}$

Where:

• $v_{\text{rms}}$ = root mean square speed
• $k_B$ = Boltzmann constant ($1.38\times 10^{-23}$)
• $T$ = absolute temperature (in Kelvin)
• $m$ = mass of a gas molecule

Since kinetic energy is proportional to temperature, the rms speed increases with higher temperatures and decreases for heavier gas molecules.

Did You Know?

The rms speed of oxygen molecules at room temperature is approximately 484 m/s, close to the speed of sound!

Real-life Application:

The rms speed of gas molecules helps in predicting the rate of diffusion of gases in the atmosphere, which is crucial in understanding gas exchange in biological systems like the human lungs.

1.3 Average Speed

The average speed is the arithmetic mean of the speeds of all gas molecules. While slightly lower than the rms speed, it gives a general idea of the motion of gas molecules. The formula is:

$v_{\text{avg}}=\sqrt{\frac{8k_{B}T}{\pi m}}$

Average speed is useful for approximating the kinetic energy of gases without needing the detailed calculations involved in rms speed.

Mnemonic:

"A Root Must Square, but on Average, 8 is Fair" – Helps recall that rms speed uses 3, while average speed uses $\frac{8}{\pi }$.

1.4 Most Probable Speed

The most probable speed is the speed at which the maximum number of gas molecules is moving. It corresponds to the peak of the Maxwell-Boltzmann distribution curve and is calculated as:

$v_{\text{mp}}=\sqrt{\frac{2k_{B}T}{m}}$

This speed is lower than both rms and average speeds, representing the most common speed among gas molecules in the system.

NEET Tip:

The relationship between the three speeds is: $v_{\text{mp}}<v_{\text{avg}}<v_{\text{rms}}$. In NEET, identify which speed the problem is asking for and use the correct formula.

1.5 Maxwell-Boltzmann Distribution of Speeds

The Maxwell-Boltzmann distribution describes how the speeds of gas molecules are distributed at a particular temperature. The curve shows:

• Most molecules move at an intermediate speed (most probable speed).
• Few molecules move very fast or very slow.

The area under the curve represents the total number of molecules, while the peak corresponds to the most probable speed. As temperature increases, the distribution shifts toward higher speeds.

NEET Problem-Solving Strategy:

When solving NEET questions, ensure you identify the type of speed being referred to in the question. Pay attention to units and ensure temperature is in Kelvin.

1.6 Visual Aids

A Maxwell-Boltzmann distribution curve could be helpful here. The graph should show the curve for different temperatures to visualize how molecular speeds change with increasing temperature. The curve's peak shifts to the right (higher speeds) as temperature rises, while the distribution flattens.

Quick Recap:

• Root Mean Square Speed: $v_{\text{rms}}=\sqrt{\frac{3k_{B}T}{m}}$
• Average Speed: $v_{\text{avg}}=\sqrt{\frac{8k_{B}T}{\pi m}}$
• Most Probable Speed: $v_{\text{mp}}=\sqrt{\frac{2k_{B}T}{m}}$
• The relationship between these speeds is: $v_{\text{mp}}<v_{\text{avg}}<v_{\text{rms}}$.
• Maxwell-Boltzmann distribution shows the spread of molecular speeds, with most molecules moving at the most probable speed.

Practice Questions:

1. Calculate the rms speed of nitrogen molecules at 300 K. (Molecular mass of $N_2$is 28 g/mol).
   Solution:
2. • Given: $T=300$ K, $m=28$ g/mol
   • Convert mass to kg: $m=28\times 10^{-3}$ kg/mol
   • Use the formula for $v_{\text{rms}}$: $v_{\text{rms}}=\sqrt{\frac{3\times 1.38\times 10^{-23}\times 300}{28\times 10^{-3}}}$ Answer: 517 m/s
2. Find the most probable speed of helium molecules at 273 K. (Molecular mass of helium is 4 g/mol).
   Solution:
3. • Given: $T=273$ K, $m=4$ g/mol
   • Convert mass to kg: $m=4\times 10^{-3}$ kg/mol
   • Use the formula for $v_{\text{mp}}$: $v_{\text{mp}}=\sqrt{\frac{2\times 1.38\times 10^{-23}\times 273}{4\times 10^{-3}}}$ Answer: 886 m/s
3. If the average speed of oxygen molecules at a certain temperature is 400 m/s, calculate the temperature of the gas.
   Solution:
4. • Given: $v^{}$