Question 207: Thermodynamics & KTG

Question

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases.
Reason (R) : Free expansion of an ideal gas is an irreversible and an adiabatic process.
In the light of the above statements, choose the correct answer from the options given below :

Accepted Answer

Let's analyze both statements step by step.
Assertion (A): "In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases."
Ã¢ÂÂÃ¢ÂÂ¢ In an adiabatic process, no heat is exchanged with the surroundings.
Ã¢ÂÂÃ¢ÂÂ¢ For a reversible adiabatic (compression) process, the relationship between temperature and volume is given by
Ã¢ÂÂÃ¢ÂÂ$$T V^{\gamma-1} = 	ext{constant},$$
Ã¢ÂÂÃ¢ÂÂwhere $$\gamma$$ is the heat capacity ratio (greater than 1).
Ã¢ÂÂÃ¢ÂÂ¢ If the volume is reduced from $$V$$ to $$V/2$$, then
Ã¢ÂÂÃ¢ÂÂ$$T_{	ext{final}} = T_{	ext{initial}} \left(\frac{V}{V/2}ight)^{\gamma-1} = T_{	ext{initial}}\,(2)^{\gamma-1}.$$
Ã¢ÂÂÃ¢ÂÂ¢ Since $$(2)^{\gamma-1} > 1,$$ the final temperature is higher than the initial temperature.
Ã¢ÂÂÃ¢ÂÂ¢ Therefore, the assertion that the temperature decreases is incorrect.
Reason (R): "Free expansion of an ideal gas is an irreversible and an adiabatic process."
Ã¢ÂÂÃ¢ÂÂ¢ In a free expansion, the gas expands into a vacuum without doing any work on the surroundings.
Ã¢ÂÂÃ¢ÂÂ¢ Since the container is insulated, no heat is exchanged, making it adiabatic.
Ã¢ÂÂÃ¢ÂÂ¢ However, the process is inherently irreversible due to the spontaneous expansion and mixing of states.
Ã¢ÂÂÃ¢ÂÂ¢ Thus, this statement is true.
Since Assertion (A) is false and Reason (R) is true (but the reason does not explain the false assertion about temperature change during compression), the correct answer is:
Option C
(A) is false but (R) is true.

Answer

Both $(\mathbf{A})$ and $(\mathbf{R})$ are true and $(\mathbf{R})$ is the correct explanation of $(\mathbf{A})$

Answer

Both (A) and (R) are true but (R) is NOT the correct explanation of (A)

Answer

(A) is false but (R) is true

Answer

(A) is true but $(\mathbf{R})$ is false

Thermodynamics & KTG

Choose the correct answer: