An ideal gas goes through a reversible isothermal expansion (solid line) followed by a reversible adiabatic expansion (dashed line). Which of the following diagram(s) closely depict(s) the entire process?
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Isothermal expansion = Temperature constant, Pressure drops, Volume increases.
Adiabatic expansion = Temperature falls, Pressure drops even faster than in the isothermal step!
Steeper slope on a P-V graph is the signature of an adiabatic curve.
Step 1: Understanding the Question:
This question asks us to identify which of the given thermodynamic graphs (P-V, T-V, P-T, P-1/V) correctly represent a two-step process: first, a reversible isothermal expansion, followed by a reversible adiabatic expansion. Step 2: Key Formula or Approach:
Let's analyze the mathematical relationships for each process:
1. Reversible Isothermal Expansion (solid line):
- $T = \text{constant}$.
- $PV = \text{constant}$ (Boyle's Law). Since it is an expansion, $V$ increases and $P$ decreases.
2. Reversible Adiabatic Expansion (dashed line):
- $PV^\gamma = \text{constant}$ (where $\gamma > 1$ is the heat capacity ratio).
- The slope of an adiabatic curve on a $P-V$ diagram is steeper than that of an isothermal curve because:
\[ \left(\frac{\partial P}{\partial V}\right)_{\text{adi}} = -\gamma \frac{P}{V} < - \frac{P}{V} = \left(\frac{\partial P}{\partial V}\right)_{\text{iso}} \]
- Since work is done at the expense of internal energy during adiabatic expansion, the temperature ($T$) decreases as volume ($V$) increases. Step 3: Detailed Explanation:
Let's test each of the four diagrams:
• Diagram (i) [P versus V]:
- The solid line (isothermal expansion) slopes downward as volume increases.
- The dashed line (adiabatic expansion) continues downward but with a clearly steeper slope ($\gamma$ times steeper).
- This is the standard, accurate representation on an indicator diagram. Thus, diagram (i) is correct.
• Diagram (ii) [T versus V]:
- For the isothermal step, $T$ is constant, so it should be represented by a horizontal line. The solid line is indeed horizontal.
- For the adiabatic expansion, temperature must decrease as the gas expands ($V$ increases). Thus, the dashed line should slope downwards.
- However, diagram (ii) shows the dashed line curving upwards, indicating a temperature increase. Thus, diagram (ii) is incorrect.
• Diagram (iii) [P versus T]:
- During the isothermal step, temperature is constant, which means the process must follow a vertical line on a P-T plot. Since it is an expansion, pressure decreases, so the solid line should go straight down. This is correctly depicted.
- During the adiabatic expansion, both pressure ($P$) and temperature ($T$) decrease. Therefore, the curve should head towards the lower-left quadrant (lower $P$ and lower $T$).
- Diagram (iii) shows the dashed line curving downwards and to the left, which correctly reflects this physical behavior. Thus, diagram (iii) is correct.
• Diagram (iv) [P versus 1/V]:
- For an isothermal process, $P = \frac{\text{constant}}{V} \propto \frac{1}{V}$.
- A plot of $P$ versus $1/V$ should yield a straight line passing through the origin. Since it is an expansion, $1/V$ decreases, so we move along this line towards the origin.
- However, diagram (iv) shows a sharp non-linear peak, which is completely incorrect for these processes. Thus, diagram (iv) is incorrect. Step 4: Final Answer:
Only diagrams (i) and (iii) are correct, which corresponds to option (A).
