The internal energy of an ideal gas

The model of an ideal gas is used in many places in the development of thermodynamics. For examples to follow, the following definition is needed An ideal gas is a gas 

whose internal energy in a closed system is a function only of temperature.  

On the molecular level, a gas with negligible intermolecular interactions fulfills both of these requirements. Kinetic-molecular theory predicts that a gas containing noninteracting molecules obeys the ideal gas equation. If intermolecular forces (the only forces that depend on intermolecular distance) are negligible, the internal energy is simply the sum of the energies of the individual molecules. These energies are independent of volume but depend on temperature. 

A gas with this second property is sometimes called a perfect gas. In Sec. 7.2 it will be shown that if a gas has the first property, it must also have the second. 

The behavior of any real gas approaches ideal-gas behavior when the gas is expanded isothermally. As the molar volume Fm becomes large and p becomes small, the average distance between molecules becomes large, and intermolecular forces become negligible. 

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Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

Thus the internal energy of an ideal gas is a function of temperature only. The variation of internal energy and enthalpy with temperature will now be calculated. 

Internal energy is stored as molecular kinetic and potential energy. The equipartition theorem can be used to estimate the translational and rotational contributions to the internal energy of an ideal gas. 

The internal energy of an ideal gas at constant temperature is independent of the volume of the gas. 

Thus, the statement is true i.e., the internal energy of an ideal gas is independent of the volume of the gas at constant temperature. In a similar way, one may prove the following statement The enthalpy of an ideal gas is independent of the pressure of the gas. 

Because the internal energy of an ideal gas is a function of temperature only, both enthalpy and Cp also depend on temperature alone. This is evident from the definition H = U + PV, or H = U + RT for an ideal gas, and from Eq. (2.21). Therefore, just as A U = j CvdT for any process involving an ideal gas, so AH = J CP dT not only for constant-pressure processes but for all finite processes. 

In the energy balance the accumulation term is zero because the internal energy of an ideal gas depends only on the temperature, and the temperature is constant. The energy transport terms involve heat and work... 

This is a natural result as the internal energy of an ideal gas depends on the temperature only, and the system is isothermal at the initial and final conditions. 

Accordingly, the internal energy of an ideal-gas mixture is made up of the pure-component internal energies,... 

The internal energy of an ideal gas depends only on its temperature. Do a first-law analysis of the following process. A sample of an ideal gas is allowed to expand at constant temperature against atmospheric pressure, (a) Does the gas do work on its surroundings (b) Is there heat exchange between the system and the surroundings If so, in which direction (c) What is AE for the gas for this process ... 

This is consistent with the law of equipartition of energy, which states that each degree of translational motion contributes kT per molecule or RT per mole to the internal energy of an ideal gas, relative to its ground-state energy. The heat capacity at constant volume Cy for molecules that cannot rotate or vibrate is... 

Joule s Law - The rate of heat production by a steady current in any part of an electrical circuit that is proportional to the resistance and to the square of the current, or, the internal energy of an ideal gas depends only on Its temperature. 

When the volume of an ideal gas is changed reversibly and isothermally, there is expansion work given by it = —nR T ln(F2/ V ) (Eq. 3.5.1). Since the internal energy of an ideal gas is constant at constant temperature, there must be heat of equal magnitude and opposite sign q = nRT ti V2/ V ). The entropy change is therefore... 

An ideal gas is cooled isothermally (at constant temperature). The internal energy of an ideal gas remains constant during an isothermal change. If is —76 J, what are At/ and w ... 

The internal energy of an ideal gas depends only on its temperature. Which statement is true of an isothermal (constant-temperature) expansion of an ideal gas against a constant external pressure Explain. 

The internal energy of an ideal gas depends only on temperature, and thus, at constant temperature, the derivative of U on the right side of Equation 3.52 is zero. Furthermore, with the ideal gas equation PV = nRT, the remaining two terms on the right side of Equation 3.52 cancel because (dV/dP)j = -nRT/P = -V/P. 

Example 11.2 produced an expression for the internal energy of an ideal gas of translating, rigidly rotating, harmonic oscillators. 

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