Before You Begin

Question #42178

Difficulty: 8/10 | 🟢 Free

The Question

A rigid, perfectly insulated container of volume $V$ is initially completely evacuated. It is connected via a valve to a large supply line containing an ideal diatomic gas maintained at a constant pressure $P_0$ and temperature $T_0$ . The valve is opened, allowing gas to flow slowly into the container until the pressure inside reaches $P_0$ , at which point the valve is closed. What is the final temperature $T_f$ of the gas inside the container?

Options

A.
T_0
B.
$\frac{5}{7} T_0$
C.
$\frac{7}{5} T_0$
D.
$\left(\frac{7}{5}\right)^{\frac{2}{7}} T_0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Molar Heat Capacity (Cv vs Cp)|Variable Mass Systems|Calculus/Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the First Law of Thermodynamics to a variable mass (open) system.

The 'Lethal Trap' is the assumption that since the container is rigid ( $dW=0$ ) and insulated ( $dQ=0$ ), the internal energy of the gas is conserved ( $C_v T_0 = C_v T_f$ ), leading to the incorrect answer $T_0$ .

Students fail to recognize that the gas in the supply line does 'flow work' ( $P_0 V$ ) to push the gas into the container.

Consequently, the energy entering the control volume must be modeled using enthalpy ( $C_p T_0$ ) rather than internal energy ( $C_v T_0$ ), resulting in a final temperature of $\gamma T_0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Molar Heat Capacity (Cv vs Cp)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free