A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, $C _{ V }=2 R$ Here, $R$ is the gas constant Initially, each side has a volume $V _{0}$ and temperature $T_{0}$ The left side has an electric heater, which is turned on at very low power to transfer heat $Q$ to the gas on the left side As a result the partition moves slowly towards the right reducing the right side volume to $V _{0} / 2$ Consequently, the gas temperatures on the left and the right sides become $T _{ L }$ and $T _{ R }$, respectively Ignore the changes in the temperatures of the cylinder, heater and the partition The value of $\frac{ T _{ R }}{ T _{0}}$ is
- $\sqrt{2}$
- $\sqrt{3}$
- $2$
- $3$
The Correct Option is A
Solution and Explanation
The problem involves a thermally insulating cylinder with a movable partition in the middle. Initially, there is one mole of ideal gas on each side of the partition, with specific heat at constant volume CV = 2R. The gas on each side has an initial volume V0 and temperature T0. The left side of the cylinder has an electric heater that is turned on at very low power to transfer heat Q to the gas on the left side. As a result, the partition moves slowly towards the right, reducing the volume on the right side to V0/2. This causes the temperatures on the left and right sides to become TL and TR, respectively. We are tasked with finding the value of TR/T0.
Step 1: Understanding the process
Since the system is thermally insulating, no heat is lost to the surroundings, and the heat Q transferred to the gas on the left side causes a change in the internal energy of the gas, which increases its temperature. The work done by the gas on the left side is used to move the partition, and as the partition moves, the volume of the gas on the right side decreases, which leads to a rise in the temperature of the gas on the right side as well.
Step 2: Applying the first law of thermodynamics
The first law of thermodynamics for the gas on the left side can be written as:
dQ = dU + dW
Where:
- dQ is the heat transferred to the gas (which is Q in this case),
- dU is the change in internal energy of the gas, and
- dW is the work done by the gas on the partition.
Step 3: Relating work and temperature change
The work done by the gas is related to the pressure and the change in volume. Since the gas on the right side is compressed, the work done by the gas can be written as:
W = P ΔV
For an ideal gas, the internal energy change dU is given by:
dU = n CV dT
Where n is the number of moles of gas and CV is the specific heat at constant volume. The number of moles of gas is 1, so the equation becomes:
dU = CV dT
Step 4: Using the adiabatic process for the right side
Since the partition moves slowly, we can assume an adiabatic process for the right side. For an adiabatic process, the relationship between temperature and volume for an ideal gas is given by:
T Vγ - 1 = constant
Where γ = CP/CV is the adiabatic index. Since the specific heat at constant volume is CV = 2R, we can use the relationship for the temperature and volume changes. Initially, the temperature is T0 on both sides, and the final temperature on the right side is related to the volume change. Given that the volume of the right side decreases to V0/2, the temperature TR increases by a factor of √2.
Final Answer:
The value of TR/T0 is √2.
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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 – An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
- Closed System – Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesn’t take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
- Open System – 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 – A process in which no heat transfer takes place.
- Isochoric Process – A thermodynamic process taking place at constant volume is known as the isochoric process.
- Isobaric Process – A process in which no change in pressure occurs.
- Isothermal Process – 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.