# What is isothermal expansion of an ideal gas?

##### 1 Answer

By definition:

**Isothermal**means the temperature does*not*change.**Expansion**means the volume has*increased*.

*Therefore, isothermal expansion is the increase in volume under constant-temperature conditions.*

In this situation, the gas **does work**, so the work is *negatively*-signed because the gas exerts energy to increase in volume.

During isothermal conditions, the **change in internal energy** *only* an ideal gas, so efficient work done is entirely transformed into efficient heat flow.

In other mathy words:

#color(blue)(w_"rev") = -int_(V_1)^(V_2) PdV#

#= -int_(V_1)^(V_2) (nRT)/VdV#

#= -nRTint_(V_1)^(V_2) 1/VdV#

#= color(blue)(-nRT ln|(V_2)/(V_1)|)# where:

#w_"rev"# is the most efficientworkpossible (reversible work) in#"J"# . It isas slow as possibleto ensure that no energy is lost to the atmosphere.#P# is thepressurein, say,#"bars"# ,#"atm"# , etc.#int_(V_1)^(V_2)dV# is theintegralfrom the initial to the final volume, which basically gives you the result of performinginfinitesimally slowwork.#dV# is the differential volume; that is, it is aninfinitesimally small changein the volume.#nRT# is aconstantfor an isothermal situation, and each variable holds the same meaning as in the ideal gas law. This can be pulled out as a coefficient in the integral.

The integral of

For expansion,

During isothermal conditions, the internal energy from the **first law of thermodynamics** is

#\mathbf(DeltaU = q_"rev" + w_"rev") = 0,# which means

#color(blue)(q_"rev" = -w_"rev")# .

As a brief comparison, **isothermal contraction** is when the volume decreases. It means work **was done on** the gas.

This makes the work *positive* because the gas absorbs the energy that was imparted into it to do work on it.