Comparing Slopes: Isotherms vs. Adiabats on a P-V Diagram

1. The Mathematical Foundation: Deriving the Slopes

To compare the curves, we need a quantitative measure of their steepness. On a P-V diagram, this is the derivative of pressure with respect to volume, $\frac{dP}{dV}$. Given your background in mathematics, deriving these slopes via implicit differentiation will be a familiar process.

Derivation of Slopes

Let's begin by watching a video that concisely derives the slopes for both isothermal and adiabatic processes and then directly compares them.

As you saw in the video, the derivations are straightforward:

1. Isothermal Process:

   • We start with the process equation: $PV=\text{constant}$.
   • Differentiating with respect to $V$, we get: $P\frac{dV}{dV}+V\frac{dP}{dV}=0⟹P+V\frac{dP}{dV}=0$.
   • Solving for the slope gives the result:
     ${\left(\frac{dP}{dV}\right)}_{\text{isothermal}}=-\frac{P}{V}$

2. Adiabatic Process:

   • We start with the process equation: $P{V}^{\gamma }=\text{constant}$.
   • Differentiating with respect to $V$, we use the product rule: $P(\gamma V^{\gamma -1})+V^{\gamma }\frac{dP}{dV}=0$.
   • Solving for the slope: $V^{\gamma }\frac{dP}{dV}=-P\gamma V^{\gamma -1}$. Dividing by $V^{\gamma }$ gives:
     ${\left(\frac{dP}{dV}\right)}_{\text{adiabatic}}=-\gamma \frac{P}{V}$

2. The Comparison: Which Curve is Steeper?

Now, we can directly compare the two expressions for the slope at any common point $(P,V)$:

${\left(\frac{dP}{dV}\right)}_{\text{adiabatic}}=\gamma \times \left(-\frac{P}{V}\right)=\gamma \times {\left(\frac{dP}{dV}\right)}_{\text{isothermal}}$

This is our central result. Since the adiabatic index $\gamma =\frac{C_p}{C_v}$ is always greater than 1 (because $C_p>C_v$), this equation tells us:

At any point of intersection on a P-V diagram, the slope of the adiabatic curve is $\gamma$ times steeper than the slope of the isothermal curve.

Both slopes are negative (as pressure decreases with increasing volume), so "steeper" means more negative.

You can review this derivation and conclusion in the following textual resource as well.

3. Visual and Physical Interpretation

The mathematical result ${\text{Slope}}_{\text{adia}}=\gamma \times {\text{Slope}}_{\text{iso}}$ has a clear graphical representation and a deep physical meaning.

Expansion Process (V increases)

As shown in the diagram above, if a gas expands from the same initial state:

• The adiabatic curve drops more sharply than the isothermal curve.
• For the same increase in volume, the final pressure in the adiabatic process is lower than in the isothermal process.

Physical Reason:

• During an isothermal expansion, the system does work on the surroundings. To keep the temperature constant ($\Delta U=0$), it must absorb an equivalent amount of heat from the surroundings ($Q=W$).
• During an adiabatic expansion, the system also does work, but it is thermally isolated ($Q=0$). The work is done entirely at the expense of its internal energy ($W=-\Delta U$). This causes the gas to cool down. According to the ideal gas law ($P=nRT/V$), this drop in temperature, in addition to the increase in volume, causes the pressure to fall more rapidly than in the isothermal case.

Compression Process (V decreases)

If we reverse the process and compress a gas from the same initial state:

• The adiabatic curve rises more sharply than the isothermal curve.
• For the same decrease in volume, the final pressure and temperature in the adiabatic process are higher than in the isothermal process.

Physical Reason:

• During an isothermal compression, work is done on the gas. To keep the temperature constant ($\Delta U=0$), this energy must be expelled from the system as heat ($Q=W$).
• During an adiabatic compression, work is done on the gas, but no heat can escape ($Q=0$). This work directly increases the internal energy of the gas ($\Delta U=-W$), causing its temperature to rise significantly. This temperature increase leads to a much higher final pressure.

The remainder of the Physics Galaxy video provides a great visual summary of these points and also discusses how the steepness depends on the atomicity of the gas ($\gamma$).

Conclusion

Today, we've dissected the relationship between isothermal and adiabatic curves on a P-V diagram. This analytical skill is crucial for interpreting thermodynamic diagrams correctly in complex JEE Advanced problems.

Key Takeaways:

• The slope of an isothermal curve is ${\left(\frac{dP}{dV}\right)}_{\text{iso}}=-\frac{P}{V}$.
• The slope of an adiabatic curve is ${\left(\frac{dP}{dV}\right)}_{\text{adia}}=-\gamma \frac{P}{V}$.
• The key relationship is: ${\text{Slope}}_{\text{adia}}=\gamma \times {\text{Slope}}_{\text{iso}}$.
• Since $\gamma >1$, the adiabatic curve is always steeper than the isothermal curve where they intersect. This is a direct consequence of the temperature change (or lack thereof) during the process.

Preview of the Next Lesson:
Isothermal ($P{V}^1=K$) and adiabatic ($P{V}^{\gamma }=K$) processes are actually specific instances of a more general process called a polytropic process, defined by $P{V}^n=K$, where $n$ is the polytropic index. In our next lesson, we will define a polytropic process and calculate the work done for a general polytropic index 'n'. This will unify our understanding of different thermodynamic processes into a single, powerful framework.