Internal pressure

The internal pressure , which depends on the cohesive forces of the particles in a gas, is a measure of the change in the internal energy of a gas when it expands or contracts at a constant temperature . It has the same unit as pressure , so the SI unit is pascal .

The internal pressure of an ideal gas is always zero.

definition

The internal pressure is defined as the partial derivative of the internal energy according to the volume at constant temperature: ${\displaystyle \pi _{T}}$ ${\displaystyle U}$ ${\displaystyle \pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}}$

With this one can write:, where is the heat capacity at constant volume and the change in internal energy at volume change and temperature change . ${\displaystyle \mathrm {d} U=\left({\frac {\partial U}{\partial T}}\right)_{V}\mathrm {d} T+\left({\frac {\partial U}{\partial V}}\right)_{T}\mathrm {d} V=C_{V}\mathrm {d} T+\pi _{T}\mathrm {d} V}$ ${\displaystyle C_{V}}$ ${\displaystyle \mathrm {d} U}$ ${\displaystyle \mathrm {d} V}$ ${\displaystyle \mathrm {d} T}$

The transformation also applies: ${\displaystyle \pi _{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}$

Derivation: According to the fundamental equation of thermodynamics , the complete differential of the internal energy for a fixed amount of substance is: ${\displaystyle \mathrm {d} U=T\mathrm {d} S-p\mathrm {d} V\ }$ If one differentiates the internal energy at constant temperature partially according to the volume, then the following applies: ${\displaystyle \left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial S}{\partial V}}\right)_{T}-p}$ With the Maxwell relation it follows: ${\displaystyle \ \left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial p}{\partial T}}\right)_{V}\ }$ ${\displaystyle \ \pi _{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p}$

Relationship with the Joule coefficient

The Joule coefficient (not to be confused with the much more common Joule-Thomson coefficient ) is defined by: ${\displaystyle \mu _{\mathrm {J} }}$ ${\displaystyle \mu _{\mathrm {JT} }}$

${\displaystyle \mu _{\mathrm {J} }=\left({\frac {\partial T}{\partial V}}\right)_{U}\ }$, i.e. the partial derivative of the temperature according to the volume (with constant internal energy).

According to the Maxwell relation # General Maxwell relation applies: ${\displaystyle \ \left({\frac {\partial T}{\partial V}}\right)_{U}=-\left({\frac {\partial T}{\partial U}}\right)_{V}\left({\frac {\partial U}{\partial V}}\right)_{T}}$

It follows: ${\displaystyle \mu _{\mathrm {J} }=-\left({\frac {\partial U}{\partial T}}\right)_{V}^{-1}\left({\frac {\partial U}{\partial V}}\right)_{T}=-{\frac {\pi _{T}}{C_{V}}}}$

If the internal pressure is then the Joule coefficient and thus the gas cools down with free expansion . ${\displaystyle \pi _{T}>0}$ ${\displaystyle \mu _{\mathrm {J} }<0}$

Internal pressure in simple gas models

The following is the general gas constant , the amount of substance and the molar volume . ${\displaystyle R}$ ${\displaystyle n}$ ${\displaystyle \textstyle V_{\mathrm {m} }={\frac {V}{n}}}$

Ideal gas

The following applies to the ideal gas model :

${\displaystyle p={\frac {nRT}{V}}}$

So is and thus: ${\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}={\frac {nR}{V}}}$

${\displaystyle \pi _{T}=T\left({\frac {\partial p}{\partial T}}\right)_{V}-p=T\cdot {\frac {nR}{V}}-{\frac {nRT}{V}}=0}$

In the case of an ideal gas, the internal pressure is always 0, the gas particles do not exert any forces on one another.

Van der Waals gas

For the Van-der-Waals gas model :

${\displaystyle p={\frac {RT}{V_{\mathrm {m} }-b}}-{\frac {a}{V_{\mathrm {m} }^{2}}}}$

with the (positive) Van-der-Waals constants and . ${\displaystyle a}$ ${\displaystyle b}$

So is and thus: ${\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}={\frac {R}{V_{\mathrm {m} }-b}}\ }$

${\displaystyle \pi _{T}=T\cdot {\frac {R}{V_{\mathrm {m} }-b}}-\left({\frac {RT}{V_{\mathrm {m} }-b}}-{\frac {a}{V_{\mathrm {m} }^{2}}}\right)={\frac {a}{V_{\mathrm {m} }^{2}}}\ }$

With Van-der-Waals gas (with ) the internal pressure is always positive and independent of the temperature, but tends towards 0. ${\displaystyle a>0}$ ${\displaystyle V\to \infty }$

Honest Kwong model

The following applies to the Redlich-Kwong model :

${\displaystyle p={\frac {RT}{V_{\mathrm {m} }-b}}-{\frac {a}{{\sqrt {T}}V_{\mathrm {m} }\left(V_{\mathrm {m} }+b\right)}}}$

So is

${\displaystyle \pi _{T}={\frac {3a}{2{\sqrt {T}}\;V_{\mathrm {m} }\left(V_{\mathrm {m} }+b\right)}}\ }$

According to this model, the cohesion between the particles becomes smaller at higher temperatures (and thus higher particle speeds).

See also

• Gay Lussac attempt

Individual evidence

1. ^ Basics of physical chemistry (W. Moore, D. Hummel, Verlag: Walter de Gruyter, 1986)
2. ↑ Das real Gas (www.uni-marburg.de, accessed on November 3, 2016)
3. ↑ ^{a b} Physical Chemistry (T. Engel, PJ Reid, Verlag Pearson Deutschland GmbH, 2006), page 77
4. ↑ CHAPTER 10 THE JOULE AND JOULE-THOMSON EXPERIMENTS (orca.phys.uvic.ca, accessed November 5, 2016)
5. ↑ Physical Chemistry (RG Mortimer, Academic Press, 2008)
6. ↑ see also formula collection (Table 12, staff.mbi-berlin.de, accessed on November 3, 2016)