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4.3.4. Isobaric Processes

Table of Contents

Constant-Pressure Change

An isobaric process is a thermodynamic process that happens at constant pressure. The word comes from "iso", meaning same, and "baric", meaning pressure. So during the whole process, the pressure does not change, even though other quantities such as volume, temperature, and internal energy may change.

In symbols, an isobaric process satisfies

$$
P = \text{constant}
$$

This kind of process is very common in everyday life. For example, if a gas in a cylinder is covered by a freely moving piston with a constant external load, the gas can expand or compress while the pressure stays approximately constant.

Pressure-Volume Behavior

On a pressure-volume graph, an isobaric process appears as a horizontal line, because the pressure remains fixed while the volume changes.

If the gas expands, the line goes to the right. If the gas is compressed, the line goes to the left.

Isobaric process on a pressure-volume graph

Because pressure is constant, calculating the work done by the gas is especially simple.

Work in an Isobaric Process

In thermodynamics, the work done by a gas during a volume change is

$$
W = \int P \, dV
$$

For an isobaric process, $P$ is constant, so this becomes

$$
W = P \Delta V = P(V_f - V_i)
$$

where $V_i$ is the initial volume and $V_f$ is the final volume.

If the gas expands, then $V_f > V_i$, so $\Delta V > 0$, and the work done by the gas is positive.

If the gas is compressed, then $V_f < V_i$, so $\Delta V < 0$, and the work done by the gas is negative.

For an isobaric process,
$$
W = P\Delta V
$$
If $\Delta V > 0$, the gas does positive work.
If $\Delta V < 0$, the gas does negative work.

The area under the horizontal line on the $P$-$V$ graph gives the work.

Work as area under an isobaric curve

Relation Between Temperature and Volume

If the gas is ideal, then the ideal gas law applies:

$$
PV = nRT
$$

Since pressure is constant in an isobaric process, we can write

$$
V \propto T
$$

for a fixed amount of gas. This means volume is directly proportional to absolute temperature.

So for two states in the same isobaric process,

$$
\frac{V_i}{T_i} = \frac{V_f}{T_f}
$$

or equivalently,

$$
\frac{V}{T} = \text{constant}
$$

This is often called Charles's law.

For an ideal gas in an isobaric process,
$$
\frac{V}{T} = \text{constant}
$$
Temperature must be measured in kelvin, not in degrees Celsius.

This means that if the temperature increases at constant pressure, the volume increases in the same ratio.

Heat and the First Law

The first law of thermodynamics states that

$$
\Delta U = Q - W
$$

For an isobaric process, the work is $W = P\Delta V$, so

$$
\Delta U = Q - P\Delta V
$$

or

$$
Q = \Delta U + P\Delta V
$$

This tells us that heat added to the gas does two things. Part of it changes the internal energy, and part of it is used to do work as the gas expands.

In an isobaric process,
$$
Q = \Delta U + P\Delta V
$$
The added heat is not used only to raise temperature. Some of it may go into expansion work.

Heat Capacity at Constant Pressure

In an isobaric process, the heat required to raise the temperature is described by the heat capacity at constant pressure, written as $C_P$.

For $n$ moles of gas,

$$
Q = n C_P \Delta T
$$

For an ideal gas, the internal energy change depends only on temperature, so

$$
\Delta U = n C_V \Delta T
$$

Since also

$$
Q = \Delta U + P\Delta V
$$

the constant-pressure heat is larger than the constant-volume heat for the same temperature increase. That is because at constant pressure, the gas expands and does work.

For an ideal gas, one important relation is

$$
C_P = C_V + R
$$

for molar heat capacities.

For an ideal gas heated at constant pressure,
$$
Q = nC_P\Delta T
$$
and
$$
C_P > C_V
$$
because some heat goes into work done during expansion.

Special Form for an Ideal Gas

Using the ideal gas law,

$$
PV = nRT
$$

and constant pressure, we get

$$
P\Delta V = nR\Delta T
$$

So the work done in an isobaric process for an ideal gas can also be written as

$$
W = nR\Delta T
$$

This is very useful when temperature change is known directly.

Combining this with $\Delta U = nC_V\Delta T$, we obtain

$$
Q = nC_P\Delta T
$$

again.

Expansion and Compression

An isobaric process can happen in two main ways.

If heat is added to a gas at constant pressure, the temperature usually rises and the gas expands.

If heat is removed from a gas at constant pressure, the temperature usually falls and the gas contracts.

The table below summarizes the signs of the main quantities for an ideal gas.

Process$\Delta T$$\Delta V$$W = P\Delta V$Typical sign of $Q$
Isobaric expansionpositivepositivepositivepositive
Isobaric compressionnegativenegativenegativenegative

A Simple Example

Suppose a gas expands at constant pressure $P = 2.0 \times 10^5 \, \text{Pa}$ from volume

$$
V_i = 1.0 \times 10^{-3} \, \text{m}^3
$$

to

$$
V_f = 3.0 \times 10^{-3} \, \text{m}^3
$$

Then

$$
\Delta V = 2.0 \times 10^{-3} \, \text{m}^3
$$

and the work done by the gas is

$$
W = P\Delta V
$$

$$
W = (2.0 \times 10^5)(2.0 \times 10^{-3}) = 4.0 \times 10^2 \, \text{J}
$$

So the gas does

$$
W = 400 \, \text{J}
$$

of work.

Physical Picture

It is helpful to imagine a gas under a movable piston. If the outside force on the piston stays the same, then the gas pressure can remain constant while the piston moves upward or downward.

Gas expanding under a movable piston at constant pressure

As heat enters the gas, the molecules move more energetically, the piston rises, and the gas volume increases. Because the external load is unchanged, the pressure stays approximately constant.

Key Results

Isobaric processes are important because they combine thermal change and mechanical work in a very clear way. The pressure stays fixed, the work is easy to calculate, and for an ideal gas the volume is directly proportional to absolute temperature.

Main formulas for an isobaric process:
$$
P = \text{constant}
$$
$$
W = P\Delta V
$$
For an ideal gas,
$$
\frac{V}{T} = \text{constant}
$$
$$
P\Delta V = nR\Delta T
$$
$$
Q = \Delta U + P\Delta V
$$
$$
Q = nC_P\Delta T
$$

These formulas make isobaric processes one of the simplest and most useful thermodynamic changes to analyze.

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