An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is n. The internal energy of one mole of the gas is un and the speed of sound in the gas is vn. At a fixed temperature and pressure, which of the following is the correct option?
The correct option is v5 > v3 and π3 > π5. To understand why, we need to analyze how speed of sound and internal energy relate to the degrees of freedom, n, of the gas molecules.
1. Internal Energy (U): For an ideal gas, the internal energy per mole in terms of degrees of freedom, n, is given by:
U = (n/2)RT
where R is the ideal gas constant and T is the temperature. This implies that internal energy increases with degrees of freedom.
2. Speed of Sound (v): The speed of sound in an ideal gas is given by:
v = β(Ξ³RT/M)
where Ξ³ is the adiabatic index (Cp/Cv), R is the ideal gas constant, T is the temperature, and M is the molar mass. Here, the adiabatic index Ξ³ is related to the degrees of freedom by:
Ξ³ = (n + 2)/n
Because the adiabatic index decreases with an increase in degrees of freedom, the speed of sound decreases as the degrees of freedom increase.
In conclusion, as the number of degrees of freedom increases, internal energy increases but the speed of sound decreases. Therefore, for a gas with degrees of freedom 5 (n=5) and another with 3 (n=3), we have:
v5 > v3 (speed of sound decreases with more degrees of freedom)
U3 > U5 (internal energy increases with more degrees of freedom)
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.
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.
