Polytropic Processes: Definition and Work Done

Hello! Welcome to your next lesson on thermodynamic processes.

In our last session, we took a deep dive into comparing isothermal and adiabatic curves, establishing that the adiabatic curve is always steeper by a factor of $γ$ . This was based on their governing equations, $P V^{1} = K$ and $P V^{γ} = K$ , respectively.

Today, we'll generalize this concept into a powerful, unifying framework. We'll explore the polytropic process, which is described by the equation $P V^{n} = constant$ , where $n$ is a constant called the polytropic index. This single equation can describe a wide range of thermodynamic changes. Our goal for this lesson is to define a polytropic process and learn how to calculate the work done for a general polytropic index 'n'. This is a critical tool for solving a broader class of problems you'll encounter in the JEE Advanced.

1. The Polytropic Process: A Unified Framework

A polytropic process is a thermodynamic process that obeys the relation:
$P V^{n} = C$
where $P$ is pressure, $V$ is volume, $n$ is the polytropic index, and $C$ is a constant.

The beauty of this definition is that it encompasses all the standard processes we've studied by assigning different values to $n$ . This makes it a very useful concept for classifying and analyzing processes.

The following resource provides a concise summary. Pay close attention to the table that connects the value of 'n' to the type of process.

Polytropic Process MCQ

This resource, 'Polytropic Process MCQ' from Testbook, clearly defines the polytropic process and its governing equation. Focus on the explanation and the summary table.

Read the 'Explanation' section and study the table under 'Additional Information'. This will quickly show you how different values of 'n' correspond to familiar processes.

As the resource shows, we can summarize the relationship as follows:

Value of n	Equation	Process
0	$P = C$	Isobaric (Constant Pressure)
1	$P V = C$	Isothermal (Constant Temperature)
$γ$	$P V^{γ} = C$	Adiabatic (Reversible)
$\infty$	$V = C$	Isochoric (Constant Volume)

Processes can also have other values of $n$ , including negative or fractional values, which you might see in more complex problems (e.g., a process where $P \propto V$ , which means $P V^{- 1} = C$ , so $n = - 1$ ).

2. Calculating Work Done in a Polytropic Process

Now for the main objective: calculating work. As always, the work done by the gas during a reversible process is given by the integral:
$W = \int_{V_{1}}^{V_{2}} P d V$

To solve this, we need to express $P$ as a function of $V$ . The polytropic relation $P V^{n} = C$ allows us to do just that: $P = C V^{- n}$ .

Let's watch a video that walks through the integration and derivation of the final work formula. Your strong calculus background will make this derivation very intuitive.

Work Expressions for POLYTROPIC Processes in 10 Minutes!

This video from 'Less Boring Lectures' provides a clear, step-by-step derivation of the work formula for a general polytropic process. It also correctly handles the special case where n=1.

Watch the segment from 05:43 to 07:39. Follow the integration of P dV and see how the final expression is simplified using the fact that P₁V₁ⁿ = P₂V₂ⁿ.

The General Formula for Work ( $n \neq = 1$ )

As derived in the video, integrating $P = C V^{- n}$ from $V_{1}$ to $V_{2}$ yields:
$W = \frac{C}{1 - n} [V_{2}^{1 - n} - V_{1}^{1 - n}]$
By substituting back $C = P_{2} V_{2}^{n}$ for the first term and $C = P_{1} V_{1}^{n}$ for the second term, we arrive at the much more practical formula:
$W = \frac{P _{2} V _{2} - P _{1} V _{1}}{1 - n} or W = \frac{P _{1} V _{1} - P _{2} V _{2}}{n - 1}$
Using the ideal gas law $P V = n_{m} R T$ (where $n_{m}$ is the number of moles), we can also express this in terms of temperature change:
$W = \frac{n _{m} R ( T _{2} - T _{1} )}{1 - n} or W = \frac{n _{m} R ( T _{1} - T _{2} )}{n - 1}$

Sign Convention Note: These formulas adhere to the physics/engineering convention where work done by the gas (expansion) is positive, and work done on the gas (compression) is negative. The formulas naturally produce the correct sign.

The Special Case ( $n = 1$ , Isothermal)

As the video notes, the general formula has a singularity at $n = 1$ . This is because the integral of $V^{- 1}$ is $ln (V)$ , not a power function. For the isothermal case ( $n = 1$ ), we recover the familiar result:
$W = n_{m} R T ln (\frac{V _{2}}{V _{1}}) = P_{1} V_{1} ln (\frac{V _{2}}{V _{1}})$

Test your understanding!

A gas undergoes a process described by the relation $P = α V^{2}$ , where $α$ is a positive constant. What is the polytropic index 'n' for this process?

Show answer

The goal is to rearrange the expression into the form $P V^{n} = constant$ .

Given: $P = α V^{2}$
Divide both sides by $V^{2}$ :
$\frac{P}{V ^{2}} = α$
$P V^{- 2} = α = constant$
Comparing this with $P V^{n} = constant$ , we can see that the polytropic index is $n = - 2$ .

3. Application: A JEE-Style Problem

Let's apply this to a problem. We will use a typical multi-part question but focus only on the part relevant to our current learning outcome: calculating work.

Consider the following problem, which is adapted from a standard JEE preparation material.

Problem:
For 1 mole of an ideal monoatomic gas, a process is described by the equation $P V^{2} = constant$ . The gas is initially at a temperature of 100 K and is expanded from a volume of 1 L to 2 L. Calculate the work done by the gas during this expansion. (You can leave your answer in terms of the gas constant R).

Solution Strategy:

Identify the polytropic index, n.
The process is $P V^{2} = constant$ , so by direct comparison with $P V^{n} = constant$ , we have $n = 2$ .
Choose the appropriate work formula.
Since we are given initial temperature and can find the final temperature, the formula $W = \frac{n _{m} R ( T _{1} - T _{2} )}{n - 1}$ is a good choice. We have $n_{m} = 1$ , $R$ , $T_{1} = 100 K$ , and $n = 2$ . We just need to find $T_{2}$ .
Find the final temperature, $T_{2}$ .
We need a relation between T and V for a polytropic process. You can derive this quickly:
From $P V = n_{m} R T$ , we have $P = \frac{n _{m} R T}{V}$ .
Substitute this into the process equation:
$(\frac{n _{m} R T}{V}) V^{n} = constant$
$T V^{n - 1} = constant$
This is a very useful relation to remember. Now, we can apply it to our states 1 and 2:
$T_{1} V_{1}^{n - 1} = T_{2} V_{2}^{n - 1}$
Plugging in the values:
$(100 K) (1 L)^{2 - 1} = T_{2} (2 L)^{2 - 1}$
$100 \cdot 1 = T_{2} \cdot 2$
$T_{2} = 50 K$
Calculate the work done.
Now we have everything we need:
$W = \frac{n _{m} R ( T _{1} - T _{2} )}{n - 1} = \frac{( 1 mol ) \cdot R \cdot ( 100 K - 50 K )}{2 - 1}$
$W = \frac{50 R}{1} = 50 R$

The work done by the gas is 50R. Since the value is positive, this confirms it's work done by the gas, which makes sense for an expansion.

The original problem also asks for heat absorbed and change in internal energy. We'll tackle those in the next lesson after we've derived the molar heat capacity for a polytropic process.

Conclusion

Today we've generalized our understanding of thermodynamic processes under the single, elegant concept of the polytropic process, $P V^{n} = C$ . This framework not only helps in classifying processes but also provides a direct method for calculating work done.

Key Takeaways:

A polytropic process is defined by $P V^{n} = constant$ , where $n$ is the polytropic index.
This single relation unifies isobaric ( $n = 0$ ), isothermal ( $n = 1$ ), adiabatic ( $n = γ$ ), and isochoric ( $n = \infty$ ) processes.
The work done in a polytropic process (for $n \neq = 1$ ) is given by:
$W = \frac{P _{1} V _{1} - P _{2} V _{2}}{n - 1} = \frac{n _{m} R ( T _{1} - T _{2} )}{n - 1}$
For the special case of an isothermal process ( $n = 1$ ), the work is $W = n_{m} R T ln (V_{2} / V_{1})$ .

Preview of the Next Lesson:
We have formulas for $Δ U$ (always $n_{m} C_{V} Δ T$ ) and now for $W$ . The missing piece in the First Law of Thermodynamics ( $Δ U = Q - W$ ) is the heat, $Q$ . In the next lesson, we will learn to determine the molar heat capacity for a given polytropic process. This will give us a direct way to calculate $Q = n_{m} C_{p o l y} Δ T$ and complete our analysis of any polytropic change.

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