Question

The speeds of 5 molecules of a gas (in arbitrary units) are as follows: \(2,3,4,5,6\), The root mean square speed for these molecules is (A) \(4.24\) (B) \(2.91\) (C) \(4.0\) (D) \(3.52\)

Short answer

The root mean square speed for these molecules is (A) \(4.24\).

Step-by-step solution

Square each speed

First, we find the square of each speed given in the problem. These are: \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\), and \(6^2= 36\).

Find the mean of the squared speeds

Now, we will find the mean of the squared speeds: \[(4 + 9 + 16 + 25 + 36) \div 5\] \[= \frac{90}{5}\] \[= 18\]

Find the square root of the mean

Finally, we will find the square root of the mean we calculated in step 2: \[\sqrt{18} \approx 4.24\]

Compare the calculated value with the options

Now, we will compare our calculated value with the given options: (A) \(4.24\) (B) \(2.91\) (C) \(4.0\) (D) \(3.52\) Our calculated value \(\approx 4.24\), which matches option (A), so the correct answer is (A) \(4.24\).

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a theoretical framework that helps us understand the behavior of gases. This theory makes several assumptions to simplify the complex motions and interactions of gas molecules. These include:

• Gas molecules are in constant and random motion.
• They collide with each other and the walls of their container, but these collisions are perfectly elastic, meaning no energy is lost in the process.
• The molecules are considered point particles with negligible volume compared to the space they occupy.
• The intermolecular forces between gas molecules are negligible except during collisions.

Based on these assumptions, the kinetic theory helps derive important properties of gases, like pressure, temperature, and molecular speed. In the context of this theory, molecular speed is crucial as it ties in with energy by relating speed to kinetic energy, which is proportional to the temperature of the gas.

Molecular Speed

Molecular speed is a pivotal concept when discussing gases, as it describes how fast the individual molecules in a gas are moving. The distribution of these speeds can be described statistically, as molecules in a gas do not move at a constant speed. The three main types of molecular speed include:

• Average Speed: This is the mean of all the speeds of the molecules.
• Most Probable Speed: This speed indicates the speed at which the largest number of molecules are moving.
• Root Mean Square Speed (RMS): This is particularly significant because it connects the macroscopic properties of the gas to its microscopic behavior, calculated as the square root of the average of the squares of the individual speeds.

To calculate RMS speed, as done in the exercise, first square each of the speeds, find the average of these squares, and then take the square root of this average. RMS provides insight into the kinetic energy of the gas since it relates directly to temperature.

Statistical Mechanics

Statistical mechanics is a branch of physics that applies statistical methods to explain and predict the behavior of systematic ensembles of particles. It is particularly useful in understanding the properties of gases due to the vast number of molecules involved.

Through statistical mechanics, one can connect microscopic behavior (such as molecular speed and energy) to macroscopic observable properties (like pressure and temperature). By using probability and statistics, this field helps to describe situations where the behavior of a huge number of particles is involved without tracking each particle individually.

In gases, statistical mechanics helps derive relationships between properties such as pressure, volume, and temperature based on molecular speed and energy distributions. This approach not only provides a deeper understanding of concepts like the root mean square speed but also elaborates on how these molecules distribute speed according to conditions like temperature and volume.