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The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.
where denotes all extensive variables of the internal energy. The internal energy is thus a first-order homogenous function. Applying Euler's homogeneous function theorem, one finds the following relation when taking only volume, number of particles, and entropy as extensive variables:
The energy that the decayed organisms originally obtained from the sun via photosynthesis is stored as chemical energy within the molecules of methane and other hydrocarbons. [8] Natural gas can be burned for heating, cooking, [9] and electricity generation. Consisting mainly of methane, natural gas is rarely used as a chemical feedstock.
The other equation of state of an ideal gas must express Joule's second law, that the internal energy of a fixed mass of ideal gas is a function only of its temperature, with = (,). For the present purposes it is convenient to postulate an exemplary version of this law by writing:
where the potentials as functions of their natural thermal and mechanical variables are the internal energy (,), enthalpy (,), Helmholtz free energy (,), and Gibbs free energy (,). The thermodynamic square can be used as a mnemonic to recall and derive these relations.
How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount, n (in moles), is equal to total mass of the gas (m) (in kilograms) divided by the molar mass, M (in kilograms per mole): =.
Each pair in the equation are known as a conjugate pair with respect to the internal energy. The intensive variables may be viewed as a generalized "force". An imbalance in the intensive variable will cause a "flow" of the extensive variable in a direction to counter the imbalance. The equation may be seen as a particular case of the chain rule.
In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. [1] [2] Most modern equations of state are formulated in the Helmholtz free energy.