Isothermal change of state

Comparison of the isothermal and isentropic changes in state in the pV diagram

The isothermal change of state is a thermodynamic change of state in which the temperature remains unchanged:

${\displaystyle T={\text{const.}}\quad \Leftrightarrow \quad T_{1}=T_{2}}$

There denote and the temperatures before and after the change of state. When a gas is compressed, the heat of compression has to be dissipated, and when it is expanded, heat has to be added ( diabatic change of state ) . This can be achieved approximately by a heat bath . ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$

Ideal gas

According to Boyle-Mariotte's law and the caloric equation of state of an ideal gas , the product of pressure and volume as well as the internal energy remain constant at constant temperature : ${\displaystyle T}$ ${\displaystyle p}$ ${\displaystyle V}$ ${\displaystyle U}$

${\displaystyle p\cdot V=n\cdot R\cdot T={\text{const.}}\quad \Leftrightarrow \quad p\sim {\frac {1}{V}}}$.

From this it follows that the pressures are inversely proportional to the corresponding volumes:

${\displaystyle {\frac {V_{1}}{V_{2}}}={\frac {p_{2}}{p_{1}}}}$

For the work done, the following applies to isothermal compression or expansion of moles of an ideal gas: ${\displaystyle \ W}$ ${\displaystyle n}$

${\displaystyle \ W=n\,R\,T_{1}\ln \left({\frac {V_{1}}{V_{2}}}\right)=n\,R\,T_{1}\ln \left({\frac {p_{2}}{p_{1}}}\right)=p_{1}\,V_{1}\ln \left({\frac {V_{1}}{V_{2}}}\right)}$,

where denotes the universal gas constant . ${\displaystyle R}$

Because is . According to the first law of thermodynamics ( ) it follows that the added or extracted heat corresponds directly to the work performed ( ). ${\displaystyle T_{2}=T_{1}}$ ${\displaystyle \Delta U=0}$ ${\displaystyle \Delta U=Q+W}$ ${\displaystyle \ Q=-W}$

See also

• Johnson-Mehl-Avrami-Kolmogorow equation ,
• Stirling engine

Web links

Commons : Isothermal processes  - collection of images, videos and audio files