# On Classical Ideal Gases

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## Abstract

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## 1. Introduction

**Figure 1.**Space-time ($z,t$) trajectory for a corpuscle of weight w bouncing off the ground (z = 0). The maximum altitude reached by the corpuscle is ${z}_{m}=E/w$, where E denotes the energy. The motion is periodic with period $\tau \left({z}_{m}\right)$, where $\tau \left(Z\right)$ denotes the corpuscle round-trip time at a distance Z from the top of the trajectory. When the altitude is restricted to h by a plate (dashed horizontal line) the motion remains periodic with a period evidently equal to: $\tau \left({z}_{m}\right)-\tau ({z}_{m}-h)$. Note that this expression holds even if the motion is not symmetric in time.

## 2. Average Force Exerted by a Corpuscle on a Piston

## 3. Internal Energy

## 4. The Energy θ is a Thermodynamic Temperature

**The Carnot cycle:**A Carnot cycle consists of two isothermal transformations at temperatures ${\theta}_{l}$ and ${\theta}_{h}$, and two intermediate reversible adiabatic transformations ($dS=0$). After a complete cycle, the entropy recovers its original value and therefore $d{S}_{l}+d{S}_{h}=0$. According to Equation (9): $-\delta {Q}_{l}={\theta}_{l}\phantom{\rule{0.166667em}{0ex}}d{S}_{l}$, $-\delta {Q}_{h}={\theta}_{h}\phantom{\rule{0.166667em}{0ex}}d{S}_{h}$ and therefore $\delta {Q}_{l}/{\theta}_{l}+\delta {Q}_{h}/{\theta}_{h}=0$. Energy conservation gives the work $\delta W$ performed over a cycle from: $\delta W+\delta {Q}_{l}+\delta {Q}_{h}=0$. The cycle efficiency is defined as the ratio of $\delta W$ and the heating $-\delta {Q}_{h}$ supplied by the hot bath. We have therefore $\eta \equiv \frac{\delta W}{-\delta {Q}_{h}}=\frac{\delta {Q}_{h}+\delta {Q}_{l}}{\delta {Q}_{h}}=1-\frac{{\theta}_{l}}{{\theta}_{h}}$, from which we conclude that θ is the “thermodynamic temperature”. Since Kelvin time, thermodynamics temperatures are strictly defined from Carnot (or other reversible) cycles efficiency. In practice, temperatures may be measured by other means and employed in other circumstances.

**Practical units:**The energy $\theta =\u2329F\u232ah$ has been defined so far only to within a multiplicative factor from dimensional considerations. This factor is fixed by agreeing that $\theta =273.16$ ${k}_{\mathrm{B}}$ exactly when the cylinder is in thermal equilibrium with water at its triple point. Here ${k}_{\mathrm{B}}=1.38066...\phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$ joules is considered as an energy unit (akin to the calorie = 4.182... joules). This manner of defining θ is equivalent to the usual one, though expressed differently. The dimensionless quantity $T\equiv \theta /{k}_{\mathrm{B}}$ is the usual unit of thermodynamic temperature, expressed in Kelvin.

## 5. Conclusions

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Arnaud, J.; Chusseau, L.; Philippe, F.
On Classical Ideal Gases. *Entropy* **2013**, *15*, 960-971.
https://doi.org/10.3390/e15030960

**AMA Style**

Arnaud J, Chusseau L, Philippe F.
On Classical Ideal Gases. *Entropy*. 2013; 15(3):960-971.
https://doi.org/10.3390/e15030960

**Chicago/Turabian Style**

Arnaud, Jacques, Laurent Chusseau, and Fabrice Philippe.
2013. "On Classical Ideal Gases" *Entropy* 15, no. 3: 960-971.
https://doi.org/10.3390/e15030960