Gay Lussac attempt

Gay-Lussac experiment: free expansion of a gas

The Gay-Lussac experiment (or Joule experiment ) is a classic experiment in thermodynamics. It consists in the free expansion of a gas into a previously evacuated volume, whereby neither work nor heat can be exchanged with the environment. In a controlled experiment, after opening a valve, a gas flows from one container into a second container until the pressure is equalized. It was first carried out in 1807 by Joseph Louis Gay-Lussac with dry air and is therefore also called the Gay-Lussac Overflow Test . In 1845 it was repeated with greater accuracy by James Prescott Joule . The result both times was that the temperature remained the same overall. Today this is explained by the fact that the air behaves here with sufficient accuracy like an ideal gas , the internal energy of which is kept constant by performing the experiment. In the case of an ideal gas, the temperature for a given amount of substance depends only on the internal energy and is in particular independent of the volume occupied. If the experiment had been carried out more precisely, however, a cooling should have been shown, because strictly speaking, air is not an ideal gas. The Gay-Lussac experiment must not be confused with the Joule-Thomson effect or the Gay-Lussac law, which also deal with the expansion of gases.

history

At the beginning of the 19th century, heat was still primarily understood as a subtle fluid that determines the temperature of a quantity of substance by the concentration in which it is present. As a result, Gay-Lussac had the expectation that the temperature would have to drop because the thermal fluid was diluted during the expansion attempt. This also applied to the half of the air volume that had remained in the previously filled container after pressure equalization, but the other half that had flowed into the previously evacuated container of the same size was heated by the same amount, so that overall no cooling entered. When in the middle of the 19th century there was increasing experience with the generation of heat through mechanical work and vice versa, doubts about the conception of heat as fluid increased. The Gay-Lussac experiment was therefore repeated in 1845 by James Prescott Joule with increased care and measurement accuracy, the two gas containers being in a thermally insulated water bath. It was confirmed that after equalizing the initial temperature differences, the same temperature was established as before the expansion. The independence of the internal energy from the volume was thus confirmed and - according to the law of Boyle and Mariotte - became the second defining property of the ideal gas.

Physical description

The internal energy of a given amount of substance changes according to the 1st law of thermodynamics around the heat and the work that are added from the environment (or, if they are negative, removed): ${\displaystyle U}$ ${\displaystyle Q}$ ${\displaystyle W}$

${\displaystyle \Delta U=Q+W}$

${\displaystyle U}$is a state variable and therefore clearly given by a function , where and the state variables are volume and temperature. A differential change is related to changes in and so: ${\displaystyle U(V,T)}$ ${\displaystyle V}$ ${\displaystyle T}$ ${\displaystyle \mathrm {d} U}$ ${\displaystyle V}$ ${\displaystyle T}$

${\displaystyle \mathrm {d} U=\left({\frac {\partial U}{\partial V}}\right)_{T}\mathrm {d} V+\left({\frac {\partial U}{\partial T}}\right)_{V}\mathrm {d} T}$

Here is the isochoric heat capacity and the internal pressure . ${\displaystyle \ C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}\ }$ ${\displaystyle \ \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}\ }$

If for a process , then for the changes in and the equation follows: ${\displaystyle \mathrm {d} U=0}$ ${\displaystyle V}$ ${\displaystyle T}$

${\displaystyle {\frac {\mathrm {d} T}{\mathrm {d} V}}=\left({\frac {\partial T}{\partial V}}\right)_{U}=-{\frac {\left({\tfrac {\partial U}{\partial V}}\right)_{T}}{\left({\tfrac {\partial U}{\partial T}}\right)_{V}}}}$

The expansion in the Gay-Lussac trial is due to such a process . That the temperature remains constant means . Hence here is ${\displaystyle Q=W=0}$ ${\displaystyle \mathrm {d} U=0}$ ${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{U}=0}$

${\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=0}$.

The internal energy of the ideal gas does not depend on the volume at a given temperature.

According to the 2nd law of thermodynamics , this property of the ideal gas can be justified from its thermal equation of state ( ). From the existence of the state variable entropy , according to the Maxwell relations , one can derive: ${\displaystyle p=nRT/V}$ ${\displaystyle S}$

${\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}$.

Because of the thermal equation of state of the ideal gas it holds that

${\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}={\frac {nR}{V}}={\frac {p}{T}}}$,

emerges as a compelling conclusion . The ideal gas is therefore already sufficiently defined by the thermal equation of state , because it already contains the condition that the internal energy is independent of the volume. ${\displaystyle \left({\tfrac {\partial U}{\partial V}}\right)_{T}=0}$ ${\displaystyle p=nRT/V}$

Real gas

A few years after Joule confirmed the Gay-Lussac experiment using the Joule-Thomson effect , Joule and Kelvin proved that air, strictly speaking, is not an ideal gas . Accordingly, if the measurements were taken more precisely, the experiments by Gay-Lussac and Joule described above should have shown a cooling. With the help of the Van der Waals equation , a thermal equation of state that corresponds better to the real gases than the equation of the ideal gas used above, the expression for the internal energy results

${\displaystyle U(T,V)=C_{V}T-{\frac {a}{V}}}$, or ${\displaystyle \mathrm {d} U=C_{V}\mathrm {d} T+{\frac {a}{V^{2}}}\mathrm {d} V}$

Therein is the heat capacity of the gas quantity at constant volume and the Van der Waals constant (which is always positive) for the attractive forces between the gas particles. Here the internal energy depends explicitly on the volume. ${\displaystyle C_{V}}$ ${\displaystyle a}$

At constant, it follows for the measurand of the experiment ${\displaystyle U_{1}=U_{2}}$

${\displaystyle C_{V}(T_{2}-T_{1})=a\left({\frac {1}{V_{2}}}-{\frac {1}{V_{1}}}\right)}$.

Expansion always draws thus cooling off after themselves: . This explicit dependence of the temperature on the volume suggests that the Joule experiments (initial pressure 22 bar at room temperature, doubling the volume) will cool the air by approx. 3.5 ° C. It was overlooked at the time, probably because the surrounding water bath had a much greater heat capacity, so that the final temperature was no longer measurably different from the initial temperature. ${\displaystyle V_{2}>V_{1}}$ ${\displaystyle T_{2}<T_{1}}$

Individual evidence

1. ^ Joseph Louis Gay-Lussac: Premier Essai pour déterminer les variations de température qu'éprouvent les gaz en changeant de densité, et considérations sur leur capacité pour le calorique. Mém. d'Arcueil 1807; reprinted again in Ernst Mach: Die Principien der Wärmelehre , Verlag von Johann Ambrosius Barth, Leipzig 1896, from p. 461. See also the explanations from p. 198
2. ↑ JP Joule: On the changes of temperature produced by the rarefaction and condensation of air , Philosophical Magazine Series 3, 26: 174 (1845), pp. 369-383, doi : 10.1080 / 14786444508645153 .
3. ↑ D. Lüdecke, C. Lüdecke: Thermodynamics - Physico-chemical basics of thermal process engineering . Springer Verlag, 2000, p. 236 ( preview ).
4. ↑ Klaus Stierstadt. Thermodynamics: From Microphysics to Macrophysics . Springer-Verlag 2010. ISBN 978-3-642-05098-5 . P. 460