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
For an ideal gas, the number of degrees of freedom (n) determines the value of Ξ³. For a monatomic gas (such as helium or argon), the number of degrees of freedom is 3, and Ξ³ is equal to \(\frac{5}{3}\).
For a diatomic gas (such as oxygen or nitrogen), the number of degrees of freedom is 5, and Ξ³ is equal to\( \frac{7}{5}\).
Now, let's analyze the options:(3) \(v_5 > v_7\) and \(π_5 < π_7\):
This option states that the speed of sound in gas 5 is greater than in gas 7, which is true because a gas with a higher number of degrees of freedom will have a higher speed of sound. It also states that the internal energy of gas 5 is less than gas 7, which is true as well because higher degrees of freedom correspond to higher internal energy. Therefore, this option is correct.
Therefore, the correct option is (3) \(v_5 > v_7\) and \(π_5 < π_7.\)
Approach Solution -2
\(U_n=\frac{1\times n\times RT}{2}\)
\(=\frac{nRT}{2}\)
\(V_n=\sqrt{\frac{\gamma RT}{M}}\)
\(=\sqrt{\frac{(1+\frac{2}{n})RT}{M}}\)
\(β\)\(U_7 > U_5\ \text{and}\ U_7 > U_6\ \text{ and}\ v_5 > v_7 \)
Top Questions on Thermodynamics
- Given are statements for certain thermodynamic variables: \begin{itemize} \item[(A)] Internal energy, volume \( V \), and mass \( M \) are extensive variables. \item[(B)] Pressure \( P \), temperature \( T \), and density \( \rho \) are intensive variables. \item[(C)] Volume \( V \), temperature \( T \), and density \( \rho \) are intensive variables. \item[(D)] Mass \( M \), temperature \( T \), and internal energy are extensive variables. \end{itemize} Choose the correct answer from the options given below:
- JEE Main - 2025
- Physics
- Thermodynamics
- For a diatomic gas, if \( \gamma_1 = \frac{C_P}{C_V} \) for rigid molecules and \( \gamma_2 = \frac{C_P}{C_V} \) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct? (where \( C_P \) and \( C_V \) are specific heats of the gas at constant pressure and volume)
- JEE Main - 2025
- Physics
- Thermodynamics
- Match List - I with List - II. List - I (Partial Derivatives) \(and\) List - II (Thermodynamic Quantity)
In the light of the above statements, choose the correct answer from the options given below:- JEE Main - 2025
- Chemistry
- Thermodynamics
- A container of fixed volume contains a gas at 27Β°C. To double the pressure of the gas, the temperature of the gas should be raised to Β°C.
- JEE Main - 2025
- Physics
- Thermodynamics
- The work done in an adiabatic change in an ideal gas depends upon:
- JEE Main - 2025
- Physics
- Thermodynamics
Questions Asked in JEE Advanced exam
- A positive, singly ionized atom of mass number \( A_M \) is accelerated from rest by the voltage \( 192 \, \text{V} \). Thereafter, it enters a rectangular region of width \( w \) with magnetic field \( \vec{B}_0 = 0.1\hat{k} \, \text{T} \). The ion finally hits a detector at the distance \( x \) below its starting trajectory. Which of the following option(s) is(are) correct? \[ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} \]
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) pointing in the \( +z \)-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height \( L \), is placed in the x-y plane with its vertex \( P \) at \( x = 0 \) in the orientation shown in the figure. At \( t = 0 \), the loop starts entering the region of the magnetic field with a uniform velocity \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf (\( \mathcal{E} \)) in the loop as a function of the distance (\( x \)) starting from \( x = 0 \)?- JEE Advanced - 2024
- Radioactivity
- A table tennis ball has radius \( \frac{3}{2} \times 10^{-2} \, \text{m} \) and mass \( \frac{22}{7} \times 10^{-3} \, \text{kg} \). It is slowly pushed down into a swimming pool to a depth \( d = 0.7 \, \text{m} \) below the water surface and then released from rest. It emerges from the water surface at speed \( v \), without getting wet, and rises to a height \( H \). Which of the following option(s) is(are) correct?\(\text{(Given: } \pi = \frac{22}{7}, g = 10 \, \text{m/s}^2, \text{density of water} = 1 \times 10^3 \, \text{kg/m}^3, \text{viscosity of water} = 1 \times 10^{-3} \, \text{Pa.s).}\)
- JEE Advanced - 2024
- Radioactivity
- An infinitely long thin wire, having a uniform charge density per unit length of \(5 \, \text{nC/m}\), is passing through a spherical shell of radius \(1 \, \text{m}\), as shown in the figure. A \(10 \, \text{nC}\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points \(P\) and \(R\), in Volt, is ______. \[ \text{(Given: In SI units } \frac{1}{4\pi\epsilon_0} = 9 \times 10^9, \ln 2 = 0.7). \] Ignore the area pierced by the wire.
- JEE Advanced - 2024
- Radioactivity
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.