# Isothermal change of state

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 ( ). ${\ displaystyle T_ {2} = T_ {1}}$ ${\ displaystyle \ Delta U = 0}$ ${\ displaystyle \ Delta U = Q + W}$ ${\ displaystyle \ Q = -W}$ 