Animated Solution for Physics - Thermodynamics: Comprehension Passage

Welcome to one of the most interesting comprehension questions from the JEE Advanced 2017 paper! This question is famous not just for its conceptual depth, but for a rather amusing printing error. In the original exam paper, all four $p$-$V$ graphs in Column 3 were printed identically!

However, a calm and analytical mind can easily bypass this visual glitch. By systematically matching the work done equations with their corresponding thermodynamic processes, we can logically deduce which graph was intended for which process. Let's break it down step by step.

The first step is to analyze the equations in Column 1. It is crucial to note that the problem defines $W$ as the work done ON the system.

1. The Isochoric Process Equation (III) states $W_{1\to 2}=0$. Work is defined as the integral of pressure with respect to volume. If the work done is zero, there is no change in volume ($\Delta V=0$). This is the hallmark of an Isochoric process. In a $p$-$V$ diagram, this is represented by a perfectly vertical line. This was intended to be Graph S.

2. The Isobaric Process Equation (II) states $W_{1\to 2}=-p{V}_2+p{V}_1=-p(V_2-V_1)$. Since the pressure $p$ is taken outside the difference, it must be constant. This defines an Isobaric process. In a $p$-$V$ diagram, constant pressure is represented by a horizontal line. This was intended to be Graph P.

3. The Isothermal Process Equation (IV) states $W_{1\to 2}=-nRT\ln \left(\frac{V_2}{V_1}\right)$. This logarithmic relationship arises from integrating $\frac{1}{V}dV$, which is only possible when temperature $T$ is constant. This is the Isothermal process. In a $p$-$V$ diagram, it forms a rectangular hyperbola. This was intended to be Graph R.

4. The Adiabatic Process Equation (I) states $W_{1\to 2}=\frac{1}{\gamma -1}(p_2{V}_2-p_1{V}_1)$. This is the classic expression for work done on a gas during an Adiabatic process, where no heat is exchanged ($p{V}^{\gamma }=\text{constant}$). The adiabatic curve is steeper than the isothermal curve, which was intended to be Graph Q.

Question 22: The Speed of Sound This question tests your historical and conceptual knowledge of physics. Sir Isaac Newton originally assumed that sound travels through air isothermally. However, Pierre-Simon Laplace corrected this by proving that the compressions and rarefactions of sound waves happen so rapidly that there is no time for heat exchange. Thus, the process is Adiabatic.

We need the combination for the Adiabatic process: Equation (I), Process (iv), and Graph (Q). This perfectly matches option (c).

Question 23: The First Law in Action We are asked to identify the process where $\Delta U=\Delta Q-p\Delta V$.

According to the First Law of Thermodynamics, the change in internal energy is the sum of heat added and work done ON the gas:

Comparing the two equations, we get $W_{\text{on gas}}=-p\Delta V$. As we deduced earlier, this is the equation for an Isobaric process. The correct combination is Equation (II), Process (iii), and Graph (P). This matches option (b).

Question 24: Finding the Perfect Match We simply need to find a valid combination among the given options. Let's evaluate option (b), which suggests (III) (ii) (S).

- (III) is $W=0$. - (ii) is the Isochoric process. - (S) is the intended vertical line graph for constant volume.

This is a flawless match! Therefore, option (b) is the correct answer.

By breaking down the problem systematically, even a confusing typo couldn't stop us from arriving at the exact right answers. Keep your concepts clear, and you'll easily navigate any curveballs the exam throws at you!