Derive the expression for internal energy of an ideal gas for monotonic gas molecules?

I need the answer as the solution of a long question.

To derive the expression for the internal energy of an ideal gas composed of monatomic gas molecules, we need to start by understanding a few fundamental concepts related to thermodynamics and kinetic theory. The internal energy of a gas is primarily a function of the kinetic energy of its molecules, especially for monatomic gases, where the energy is solely translational. Let's break this down step by step.

Understanding Internal Energy

Internal energy (U) is the total energy contained within a system. For an ideal gas, this energy is related to the motion of its molecules. In the case of monatomic gases, the molecules possess kinetic energy due to their translational motion, and there are no other forms of energy (like rotational or vibrational) to consider.

Kinetic Energy of Monatomic Gas Molecules

The kinetic energy (KE) of a single molecule can be expressed using the formula:

• KE = (1/2) mv²

Where:

• m is the mass of the molecule
• v is the velocity of the molecule

Average Kinetic Energy

In a gas, molecules are in constant random motion, and their velocities vary. Therefore, we need to consider the average kinetic energy of the molecules. The average kinetic energy per molecule in a monatomic ideal gas is given by:

• KE_avg = (3/2) kT

Here, k is the Boltzmann constant, and T is the absolute temperature in Kelvin. This relationship arises from statistical mechanics and reflects how the energy of the gas molecules is distributed at a given temperature.

Total Internal Energy of the Gas

To find the total internal energy of the gas, we multiply the average kinetic energy per molecule by the total number of molecules (N) in the gas:

• U = N * KE_avg

Substituting the expression for average kinetic energy, we have:

• U = N * (3/2) kT

Relating Number of Molecules to Moles

We can relate the number of molecules to the number of moles (n) using Avogadro's number (N_A), which is approximately \(6.022 \times 10^{23}\) molecules/mol:

• N = n * N_A

Substituting this into our equation for internal energy gives:

• U = n * N_A * (3/2) kT

Using the Ideal Gas Constant

Recognizing that \(k = R/N_A\), where R is the ideal gas constant (approximately \(8.314 \, J/(mol \cdot K)\)), we can rewrite the internal energy expression:

• U = n * N_A * (3/2) (R/N_A) T

After simplifying, we find:

• U = (3/2) nRT

Final Expression for Internal Energy

Thus, the expression for the internal energy of an ideal gas with monatomic molecules is:

• U = (3/2) nRT

This equation indicates that the internal energy of a monatomic ideal gas is directly proportional to the number of moles and the temperature of the gas. It highlights the relationship between thermal energy and molecular motion, which is fundamental in thermodynamics.

Summary

In summary, we derived the expression for the internal energy of a monatomic ideal gas by considering the average kinetic energy of the gas molecules, relating it to the total number of molecules, and then expressing it in terms of moles and temperature. This derivation not only illustrates the principles of kinetic theory but also emphasizes the simplicity of the internal energy concept for ideal gases.