The following lines are from my book.
The gas is heated and allowed to expand at such a rate that the fall in temperature due to expansion is less than the rise in temperature due to the heat supplied.
It means that the heat supplied to the gas is used in its expansion.
But why is there a fall in the temperature of the gas due to expansion, when heat is already supplied?
It means that the heat supplied to the gas is used in its expansion.
From the text you quoted, it seems that part of the heat energy supplied to the gas is used for its expansion. This is what you usually see around you: when you heat a gas, that gas can expand a little but this does not mean that you don't manage to heat it.
Intuitively, For a gas,if you apply heat to the container of gas the kinetic energies of the molecules or atoms increase,means heat added is used in increasing the kinetic energies of the molecules. As we know ,temperature of a gas depends on how fast the molecules of gas moving or vibrating ,so on heating temperature of the gas increases.
Now these molecules also push (or collide with the walls)the piston of the container (or cylinder) and transfer a part of their kinetic energies to the pinton and the piston moves up.(means the gas expand in other words). As we know energy need to be conserved. As the pinton gains the kinetic energy ,the kinetic energies of the molecules decrease. So as I said above the temperature depends on how fast the particles are moving so temperature of the gas decreases.
When a gas expands, it has do pressure-volume work against the piston, atmosphere etc. to make its room to occupy. Now, as the system spends energy to work, part of its kinetic energy decreases. Since, temperature is directly proportional to kinetic energy, the system's temperature decreases.
An example: Much of chemistry takes place in vessels that are open to the atmosphere, and subjected to constant pressure, not constant volume in a rigid,sealed container. In general,when a change takes place in a system open to the atmosphere, its volume changes. For example, the thermal decomposition of of $1.0 \text{mol} \mathbf{CaC{0_3}}$ at $1 \text{bar}$ results in an increase in volume of nearly $90~\text{dm^3}$ at $800^\circ ~\text{C}$ on account of the carbon dioxide gas produced. To create this large volume for the carbondioxide to occupy, the surrounding atmosphere must be pushed back. That is, the system must perform expansion work. Therefore, although a certain quantity of heat may be supplied to bring about the endothermic decomposition, the increase in internal energy of the system is not equal to the energy supplied as heat because some energy has been used to do work of expansion. In other words, because the volume has increased, some of the heat supplied to the system has leaked back into the surroundings as work.
source: Elements of Physical Chemistry by Peter Atkins, Julio De Paula.
In usual laboratory situations temperature will not drop in isobaric expansion -- i.e. expansion with constant pressure. However, you will have to add heat to cause the expansion and push back the piston. Usually the temperature will also increase, it is reflected by the $c_p$ quantity (specific heat capacity at constant pressure), $c_p=\frac{Q}{\Delta T}$. $c_p$ is higher than $c_v$ because the expansion will take up energy.
Usually $c_p$ is positive, i.e. addition of heat will increase temperature, but there are thermodynamic systems that will react to additional energy by a drop in temperature and pressure. [wikipedia.org: Heat capacity, Negative heat capacity (stars)].
