Internal pressure

Thermodynamics
Branches Classical · Statistical · Chemical Equilibrium / Non-equilibrium Thermofluids
Laws Zeroth · First · Second · Third
Systems State: Equation of state Ideal gas · Real gas Phase of matter · Equilibrium Control volume · Instruments Processes: Isobaric · Isochoric · Isothermal Adiabatic · Isentropic · Isenthalpic Quasistatic · Polytropic Free expansion Reversibility · Irreversibility Endoreversibility Cycles: Heat engines · Heat pumps Thermal efficiency
System properties Property diagrams Intensive and extensive properties State functions: Temperature / Entropy (intro.) † Pressure / Volume † Chemical potential / Particle no. † († Conjugate variables) Vapor quality Reduced properties Process functions: Work · Heat
Material properties Specific heat capacity  Compressibility  Thermal expansion  Property database
Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Table of thermodynamic equations
Potentials Free energy · Free entropy Internal energy Enthalpy Helmholtz free energy Gibbs free energy
History and culture Philosophy: Entropy and time · Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics History: General · Heat · Entropy · Gas laws Perpetual motion Theories: Caloric theory · Vis viva Theory of heat Mechanical equivalent of heat Motive power Publications: "An Experimental Enquiry Concerning ... Heat" "On the Equilibrium of Heterogeneous Substances" "Reflections on the Motive Power of Fire" Timelines of: Thermodynamics · Heat engines Art: Maxwell's thermodynamic surface Education: Entropy as energy dispersal
Scientists Daniel Bernoulli Sadi Carnot Benoît Paul Émile Clapeyron Rudolf Clausius Hermann von Helmholtz Constantin Carathéodory Pierre Duhem Josiah Willard Gibbs James Prescott Joule James Clerk Maxwell Julius Robert von Mayer William Rankine John Smeaton Georg Ernst Stahl Benjamin Thompson William Thomson, 1st Baron Kelvin John James Waterston

Internal pressure is a measure of how the internal energy of a system changes when it expands or contracts at constant temperature. It has the same dimensions as pressure, the SI unit of which is 1 pascal.

Internal pressure is usually given the symbol . It is defined as a partial derivative of internal energy with respect to volume at constant temperature:

Thermodynamic equation of state

Internal pressure can be expressed in terms of temperature, pressure and their mutual dependence:

This equation is known as the thermodynamic equation of state for it expresses pressure in terms of thermodynamic properties of the system.

Derivation of the thermodynamic equation of state

Internal energy is a state function, therefore its differential is exact. We can take it to be a function of other state functions, namely entropy S and volume V: . Diving this equation by at constant temperature gives: Comparing the above differential with the fundamental thermodynamic equation gives and . This, and the definition of internal pressure, leads to One of the Maxwell relations states that Inserting this into the above relation completes the proof.

Perfect gas

In a perfect gas, there are no potential energy interactions between the particles, so any change in the internal energy of the gas is directly proportional to the change in the kinetic energy of its constituent species and therefore also to the change in temperature:

.

The internal pressure is taken to be at constant temperature, therefore

, which implies and finally ,

i.e. the internal energy of a perfect gas is independent of the volume it occupies. The above relation can be used as a definition of a perfect gas.

The relation can be proved without the need to invoke any molecular arguments. It follows directly from the thermodynamic equation of state if we use the ideal gas law .

Real gases

Real gases have non-zero internal pressures because their internal energy changes as the gases expand isothermally - it can increase on expansion (, signifying presence of dominant attractive forces between the particles of the gas) or decrease (,dominant repulsion).

In the limit of infinite volume these internal pressures reach the value of zero:

,

corresponding to the fact that all real gases can be approximated to be perfect in the limit of a suitably large volume. The above considerations are summarized on the graph on the right.

If a real gas can be described by the van der Waals equation of state

it follows from the thermodynamic equation of state that

Since the paramater is always positive, so is its internal pressure: internal energy of a van der Waals gas always increases when it expands isothermally.

The Joule experiment

James Joule tried to measure the internal pressure of air in his expansion experiment by isothermally pumping high pressure air from one metal vessel into another evacuated one. The water bath in which the system was immersed did not change its temperature, signifying that that no change in the internal energy occurred, the internal pressure of the air was equal to zero and the air was a perfect gas. The actual deviations from the perfect behaviour were not observed since they are very small and the specific heat capacity of water is relatively high.

References

Peter Atkins and Julio de Paula, Physical Chemistry 8th edition, pp. 60–61