In any spontaneous or realistic process, we always have [5] [18]

For a matter in the same phase state, the internal energy change has to be equal to zero in any isothermal process. The Equation (2a) can be derived from the Clausius inequality. The real gases that obey the Equation (2a) had been proven [5]. The ideal gas, real gases, liquids, and solids all obey the Equation (2a) too. According to the Equation (2a), in the isothermal process we have
$\text{d}U={C}_{V}\text{d}T=0$. So that, U has to be a constant (because the differential of constant is equal to zero in mathematics). Where,
$U={U}_{0}+{\displaystyle {\int}_{i}^{f}{C}_{V}\text{d}T}$. The U_{0} values for identical matter in the solid, liquid, and gas status are different at the same temperature. U_{0} hasn’t any relationship with C_{V}.

If a matter appears phase transition in the isothermal process, the following equation will be given by

$\Delta U={\displaystyle {\int}_{f}{C}_{V}\text{d}T}-{\displaystyle {\int}_{i}{C}_{V}\text{d}T}=C\left(f\right)-C\left(i\right)$,(2b)

where, i and f express the initial state and final state of a matter in the phase transition, respectively. C(i) and C(f) are all constants.

When the matters occur the variety at the chemical reactions or nuclear reactions in the isothermal process, we can obtain the Equation (2b) too, but i and f express the reactants and products respectively. C(i) and C(f) are all constants too. Hence, ΔU has to be equal to constant or zero in any isothermal process. Then, in the isothermal process, the following equation can be given by

$C\left(f\right)-C\left(i\right)=\Delta H-p\Delta V$.(2c)

In the extended or modified Bernoulli equations,
$\Delta {E}_{kin}$ and
$\Delta {{E}^{\prime}}_{kin}$ are all equal to
${\int}_{i}^{f}m\upsilon \text{d}\upsilon$ or
$\underset{i}{\overset{f}{\sum}}m\upsilon \text{d}\upsilon$,
${\int}_{i}^{f}V\text{d}p}={\displaystyle \underset{i}{\overset{f}{\sum}}V\text{d}p$,
${\int}_{i}^{f}Sh\text{d}p}={\displaystyle \underset{i}{\overset{f}{\sum}}Sh\text{d}p$ for the constant volume process,
$\int}_{i}^{f}mg\text{d}{H}_{e}}={\displaystyle \underset{i}{\overset{f}{\sum}}mg\text{d}{H}_{e$,
$\int}_{i}^{f}mg\text{d}{H}_{f}}={\displaystyle \underset{i}{\overset{f}{\sum}}mg\text{d}{H}_{f$, and
${\int}_{i}^{f}mg\text{d}h}={\displaystyle \underset{i}{\overset{f}{\sum}}mg\text{d}h}\approx {\displaystyle {\int}_{i}^{f}{G}_{0}\frac{m\cdot {m}_{0}}{{\left({r}_{0}+h\right)}^{2}}\text{d}h}={\displaystyle \underset{i}{\overset{f}{\sum}}{G}_{0}\frac{m\cdot {m}_{0}}{{\left({r}_{0}+h\right)}^{2}}\text{d}h$, where, υ expresses velocity, m_{0} is the mass of earth, r_{0} is the radius of earth, G_{0} is the gravitational constant, g is the acceleration of gravity, h is the elevation. ∑ is summation symbol. Attentively, g isn’t constant.

According to Boltzmann density distribution equation for the atmosphere, we can obtain an approximate equation as
$h=-\frac{kT}{Mg}\mathrm{ln}\frac{p}{{p}_{0}}$, where, M is mass of air molecule, k is Boltzmann constant, p is equal to p_{0} when h = 0. Attentively, ln expresses natural logarithm symbol. The right Boltzmann density distribution equation for the atmosphere should be
$p={p}_{0}{\text{e}}^{-\frac{{G}_{0}M\cdot {m}_{0}h}{kT\left({r}_{0}+h\right){r}_{0}}}$.