An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is π. The internal energy of one mole of the gas is ππ and the speed of sound in the gas is vπ. At a fixed temperature and pressure, which of the following is the correct option?
- v3 < v6 and π3 > π6
- v5 > v3 and π3 > π5
- v5 > v7 and π5 < π7
- v6 < v7 and π6 < π7
The Correct Option is C
Approach Solution - 1
Gas Properties and Sound Speed Analysis
For an ideal gas, the number of degrees of freedom (\(n\)) determines the value of \(\gamma\). Here's how it works for different types of gases:
For a **monatomic gas** (such as helium or argon), the number of degrees of freedom is 3, and \(\gamma\) is equal to: \(\frac{5}{3}\).
For a **diatomic gas** (such as oxygen or nitrogen), the number of degrees of freedom is 5, and \(\gamma\) is equal to: \(\frac{7}{5}\).
Analyzing the Options:
Option (3) states the following:
\(v_5 > v_7\)and \(U_5 < U_7\):
This option suggests: - The speed of sound in gas 5 is greater than in gas 7. This is true because a gas with a higher number of degrees of freedom will have a higher speed of sound. The speed of sound in a gas is related to \(\gamma\) and the degrees of freedom; a higher number of degrees of freedom results in a higher speed of sound.
Additionally, it states that the internal energy of gas 5 is less than that of gas 7. This is also true because a gas with more degrees of freedom has a higher internal energy. The internal energy of an ideal gas is proportional to the number of degrees of freedom.
Therefore, this option is correct: \(v_5 > v_7\)and \(U_5 < U_7 \\)
Conclusion:
The correct option is (3): \(v_5 > v_7\)and \(U_5 < U_7 \\)
Approach Solution -2
\(U_n=\frac{1\times n\times RT}{2}\)
\(=\frac{nRT}{2}\)
\(V_n=\sqrt{\frac{\gamma RT}{M}}\)
\(=\sqrt{\frac{(1+\frac{2}{n})RT}{M}}\)
\(β\)\(U_7 > U_5\ \text{and}\ U_7 > U_6\ \text{ and}\ v_5 > v_7 \)
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Concepts Used:
Thermodynamics
Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.
Important Terms
System
A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. The system boundary may be real or imaginary, fixed or deformable.
There are three types of systems:
- Isolated System β An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
- Closed System β Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesnβt take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
- Open System β In an open system, the mass and energy both may be transferred between the system and surroundings. A steam turbine is an example of an open system.
Thermodynamic Process
A system undergoes a thermodynamic process when there is some energetic change within the system that is associated with changes in pressure, volume and internal energy.
There are four types of thermodynamic process that have their unique properties, and they are:
- Adiabatic Process β A process in which no heat transfer takes place.
- Isochoric Process β A thermodynamic process taking place at constant volume is known as the isochoric process.
- Isobaric Process β A process in which no change in pressure occurs.
- Isothermal Process β A process in which no change in temperature occurs.
Laws of Thermodynamics
Zeroth Law of Thermodynamics
The Zeroth law of thermodynamics states that if two bodies are individually in equilibrium with a separate third body, then the first two bodies are also in thermal equilibrium with each other.
First Law of Thermodynamics
The First law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter transfer, and relating them to a function of a body's state, called internal energy.
Second Law of Thermodynamics
The Second law of thermodynamics is a physical law of thermodynamics about heat and loss in its conversion.
Third Law of Thermodynamics
Third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: The entropy of a system approaches a constant value when its temperature approaches absolute zero.