Which one is the correct option for the two different thermodynamic processes ?
(a) C and A
(b) C an

Q: Why is graph A incorrect? The adiabatic curve is steeper than isothermal!

While it is true that the adiabatic curve is steeper than the isothermal curve on a $p-V$ diagram, graph A shows the adiabatic process as a perfectly vertical line. A vertical line on a $p-V$ graph means the volume is constant, which represents an isochoric process, not an adiabatic one.

Q: Why is the isothermal process a vertical line in graphs C and D?

In both graphs C and D, the x-axis represents Temperature ($T$). Since an isothermal process occurs at a constant temperature, the value of $T$ does not change. This is mathematically represented by a vertical line at that specific temperature value.

Q: How do we know the shape of the adiabatic curve on a $V-T$ graph?

For an adiabatic process, $TV^{\gamma-1} = \text{constant}$. Rearranging this gives $V \propto T^{-\frac{1}{\gamma-1}}$. Since $\gamma > 1$ (e.g., 1.4 for diatomic gas), the exponent is negative. This means as temperature increases, volume decreases, resulting in a downward-sloping curve as seen in graph C.

Q: Why does the adiabatic curve in graph D go downwards and to the left?

In graph D, the adiabatic relation is $p^{1-\gamma}T^\gamma = \text{constant}$, which simplifies to $p \propto T^{\frac{\gamma}{\gamma-1}}$. Since $\frac{\gamma}{\gamma-1} > 1$, pressure and temperature are directly related. If the gas expands adiabatically, both temperature and pressure decrease, which corresponds to moving down and to the left along the curve.

Q: Can an adiabatic process ever be a straight line?

An adiabatic process is only a straight line if plotted on axes that linearize its equation, such as $\ln(p)$ vs $\ln(V)$. On standard linear $p-V$, $p-T$, or $V-T$ axes, it will always be a curve because the relationships involve exponents like $\gamma$.

Question

Which one is the correct option for the two different thermodynamic processes ?

Options:

(A) C and A

(B) C and D

(C) Only A

(D) B and C

Sigma Percentile · Accepted Answer

The world of thermodynamics is beautifully visual. When we plot the state of a gas on a graph, we aren't just drawing lines; we are tracing the exact physical journey of the molecules as they expand, compress, heat up, and cool down.

In this problem, we are presented with four different graphs and asked to identify which ones correctly represent **isothermal** and **adiabatic** processes.

This is a classic test of your graphical intuition. It's very easy to memorize that "an adiabatic curve is steeper than an isothermal curve" on a $p-V$ diagram, but what happens when the axes change? Let's dive in and decode each graph one by one!

### Evaluating Graph A: The $p-V$ Diagram

Graph A plots pressure ($p$) against volume ($V$).

For an **isothermal process**, the temperature $T$ is constant. According to the ideal gas law, $pV = 	ext{constant}$, which graphs as a rectangular hyperbola.

For an **adiabatic process**, there is no heat exchange, and the relationship is $pV^\gamma = 	ext{constant}$. Because $\gamma > 1$, this curve drops off more sharply than the isothermal one.

However, look closely at Graph A. It shows the adiabatic process as a perfectly **vertical line**. A vertical line on a $p-V$ graph means that the volume is not changing at all. This represents an **isochoric process**, not an adiabatic one!

Therefore, Graph A is fundamentally incorrect.

### Evaluating Graph B: The $p-T$ Diagram

Graph B shifts our perspective to a $p-T$ diagram, plotting pressure against temperature.

An **isothermal process** is defined by a constant temperature. On a graph where temperature is the x-axis, a constant temperature must be represented by a **vertical line**.

But what does Graph B show? It depicts the isothermal process as a **horizontal line**. A horizontal line on a $p-T$ graph means the pressure is constant, which is the definition of an **isobaric process**.

Because of this glaring error, we can immediately eliminate Graph B.

### Evaluating Graph C: The $V-T$ Diagram

Now we look at Graph C, which plots volume ($V$) against temperature ($T$).

First, let's check the isothermal process. As we established, constant temperature on a graph with a $T$-axis must be a vertical line. Graph C correctly shows the isothermal process as a vertical line.

Next, let's derive the shape of the adiabatic curve. The adiabatic relation involving volume and temperature is:
$$TV^{\gamma-1} = 	ext{constant}$$

If we rearrange this to solve for volume, we get:
$$V \propto T^{-\frac{1}{\gamma-1}}$$

Since $\gamma$ (the ratio of specific heats) is always greater than 1, the exponent $-\frac{1}{\gamma-1}$ is strictly negative. This mathematical relationship tells us a physical truth: in an adiabatic process, **as temperature increases, volume must decrease**.

Looking at the curve in Graph C, it slopes downwards to the right, perfectly illustrating that volume decreases as temperature increases.

Thus, Graph C is a correct representation!

### Evaluating Graph D: The $p-T$ Diagram

Finally, let's analyze Graph D, another $p-T$ diagram.

Once again, the isothermal process is correctly shown as a vertical line, representing constant temperature.

For the adiabatic process, we need the relation between pressure and temperature:
$$p^{1-\gamma}T^\gamma = 	ext{constant}$$

Rearranging this to see how pressure depends on temperature, we get:
$$p \propto T^{\frac{\gamma}{\gamma-1}}$$

Because $\gamma > 1$, the fraction $\frac{\gamma}{\gamma-1}$ is positive and greater than 1. This means that **pressure and temperature are directly related**—as temperature increases, pressure increases. Furthermore, because the exponent is greater than 1, the curve will bend upwards (it is concave up).

Graph D shows exactly this behavior. The curve moves downwards and to the left, meaning that as temperature decreases, pressure decreases along a concave path.

Therefore, Graph D is also a correct representation!

### The Final Verdict

After carefully translating the physical laws into graphical shapes for each set of axes, we have found that both **Graph C** and **Graph D** are correct.

Looking at our options, option (b) states "C and D". This is our final answer.

The key takeaway here is to never blindly trust the shape of a curve without first checking the axes. A hyperbola on one graph might be a completely different curve on another!

Animated Solution for Physics - Thermodynamics: Which one is the correct option for the two different thermodynamic processes ?

Select Answer:

Visualized Solution

$Analyzing the Graphs$

$Graph A: p - V Diagram$

$Graph B: p - T Diagram$

$Graph C: V - T Diagram$

$Graph D: p - T Diagram$

$Final Conclusion$

The Sigma Insight: Thermodynamic Processes

Evaluating Graph A

Evaluating Graph B

Evaluating Graph C

Evaluating Graph D

The Final Verdict

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