The degree of freedom of an ideal gas is n. The internal energy of 1 mole of the gas is Un and the speed of sound n the gas is Vn. At a fixed temperature and pressure, which of the following is correct?
- V5>V7 and U5>U7
- V5>V3 and U3>U5
- V3<V6 and U3>U6
- V6>V7 and U6<U7
The Correct Option is D
Solution and Explanation
Sound Velocity and Internal Energy: Explanation
The correct option is (D): \( V_6 > V_7 \) and \( U_6 < U_7 \).
The formulas provided for the sound velocity and internal energy are:
\(V = \sqrt{\frac{\gamma RT}{M_0}}\)
where \( \gamma = 1 + \frac{2}{n} \),
and
\(U = n' \frac{n}{2RT}\)
Explanation:
Let's break down the formulas and the effect of the number of degrees of freedom (\(n\)):
1. Sound Velocity Equation:
The formula for sound velocity \(V\) is:
\(V = \sqrt{\frac{\gamma RT}{M_0}}\)
Here, \(V\) is the sound velocity, \(R\) is the gas constant, \(T\) is the temperature, and \(M_0\) is the molar mass.
As \(n\) (the number of degrees of freedom) increases, \(\gamma\) decreases because:
\(\gamma = 1 + \frac{2}{n}\)
Since \(\gamma\) appears in the numerator of the sound velocity equation, a decrease in \(\gamma\) leads to a decrease in the sound velocity \(V\).
2. Internal Energy Equation:
The formula for internal energy \(U\) is:
\(U = n' \frac{n}{2RT}\)
Here, \(n'\) is a constant and \(n\) represents the number of degrees of freedom. As \(n\) increases, the internal energy \(U\) increases because \(U\) is directly proportional to \(n\).
Conclusion:
Thus, when \(n\) increases:
- \(\gamma\) decreases, which results in a decrease in the sound velocity (\(V\)).
- The internal energy \(U\) increases because it is directly proportional to \(n\).
Therefore, the correct option (D) is confirmed: \( V_6 > V_7 \) and \( U_6 < U_7 \).
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Concepts Used:
kinetic theory
Kinetic theory is a fundamental concept in physics that provides a microscopic explanation of the behavior of matter in terms of the motion of its constituent particles. It describes the relationship between the microscopic properties of particles, such as their motion and interactions, and the macroscopic properties of matter, such as temperature and pressure.
The key postulates of the kinetic theory are as follows:
Matter is composed of a large number of particles, such as atoms or molecules, that are in constant motion. These particles possess kinetic energy due to their motion.
The particles in a substance undergo random motion and collisions with each other and with the walls of their container. These collisions are elastic, meaning there is no loss of kinetic energy during the collision.
The volume occupied by the particles themselves is negligible compared to the total volume of the substance.
The particles experience forces of attraction or repulsion between each other, depending on their nature and distance.
Based on these postulates, the kinetic theory allows us to explain several macroscopic properties of matter. For example:
Temperature: The temperature of a substance is related to the average kinetic energy of its particles. Higher temperature corresponds to greater average kinetic energy.
Pressure: The pressure exerted by a gas is a result of the collisions of its particles with the walls of the container. The frequency and force of these collisions determine the pressure.
Diffusion: The process of diffusion, where particles spread out from an area of high concentration to an area of low concentration, can be explained by the random motion and collisions of particles.
Thermal expansion: When a substance is heated, its particles gain kinetic energy and move more vigorously, causing the substance to expand.
The kinetic theory is widely used in various fields, including thermodynamics, fluid dynamics, and statistical mechanics. It provides a foundation for understanding the behavior of gases, liquids, and solids, and it helps us develop models and theories to explain and predict their properties.