Molar internal energy

G = Gibbs molar free energy S = molar entropy F = Helmholtz free molar energy H = molar enthalpy U = molar internal energy... 

A linear molecule, such as any diatomic molecule, carbon dioxide, and ethyne (acetylene, HC=CH), can rotate about two axes perpendicular to the line of atoms, and so it has two rotational modes of motion. Its average rotational energy is therefore 2 X jkT = kT, and the contribution to the molar internal energy is NA times this value ... 

A nonlinear molecule, such as water, methane, or benzene, can rotate about any of three perpendicular axes, and so it has three rotational modes of motion. The average rotational energy of such a molecule is therefore 3 X jkT = ]kT. The contribution of rotation to the molar internal energy of a gas of nonlinear molecules is therefore... 

We have shown that the contribution to the molar internal energy of a monatomic ideal gas (such as argon) that arises from molecular motion is jRT. We can conclude that if the gas is heated through AT, then the change in its molar internal energy, AUm, is Al/m = fTAT For instance, if the gas is heated from 20.°C to 1()0.°C (so AT = +80. K), then its molar internal energy increases by 1.0 kj-mol. ... 

We can see how the values of heat capacities depend on molecular properties by using the relations in Section 6.7. We start with a simple system, a monatomic ideal gas such as argon. We saw in Section 6.7 that the molar internal energy of a monatomic ideal gas at a temperature T is RT and that the change in molar internal energy when the temperature is changed by AT is A(Jm = jRAT. It follows from Eq. 12a that the molar heat capacity at constant volume is... 

The molar heat capacities of gases composed of molecules (as distinct from atoms) are Higher than those of monatomic gases because the molecules can store energy as rotational kinetic energy as well as translational kinetic energy. We saw in Section 6.7 that the rotational motion of linear molecules contributes another RT to the molar internal energy ... 

The molar internal energy of a monoatomic ideal gas is therefore... 

The internal energy is, as indicated above, connected to the number of degrees of freedom of the molecule that is the number of squared terms in the Hamiltonian function or the number of independent coordinates needed to describe the motion of the system. Each degree of freedom contributes jRT to the molar internal energy in the classical limit, e.g. at sufficiently high temperatures. A monoatomic gas has three translational degrees of freedom and hence, as shown above, Um =3/2RT andCy m =3/2R. 

U = molar internal energy = molar volume T = absolute temperature.) This small expansion does not necessarily disrupt all the intermolecnlar solvent-solvent interactions. 

Thus, each vanishing frequency contributes a factor of RT to the molar internal energy (and thus the enthalpy). 

The molar enthalpy and the molar internal energy of bonding, Hab and Uab, are related to the bond energies Eab at constant pressure and at constant volume, respectively. 

Because RT = 2.48 kj mol 1 at 25°C, the translational motion of gas molecules contributes about 3.72 kj-mol-1 to the molar internal energy of the sample at 25°C. Apart from the energies of the atoms themselves, this is the only motional contribution to the internal energy of a monatomic gas, such as argon or any other noble gas. 

Sm is the molar entropy, Vm the molar volume of the gas at a distance x, and Um is the molar internal energy. Integrating the left side from infinite distance to a distance x leads to... 

Internal energy is an extensive property of a system. If we double the size of a system, keeping intensive variables such as temperature and pressure constant, we double the system s internal energy. If we divide the internal energy of a system by the number of moles in the system, we obtain the molar internal energy, Um = VIn, which is an intensive quantity. Other molar properties, such as the molar volume, are also indicated by the subscript m. 

Does Eq. (8) imply that for a single-component system, the chemical potential is the molar internal energy as well as the molar Gibbs free energy ... 

Here, M, may represent the partial molar internal energy Uit the partial molar enthalpy H(, the partial molar entropy Sif the partial molar Gibbs energy Gt, etc. Comparison of Eq. (10.1) with Eq. (11.2) written for the Gibbs energy shows that the chemical potential and the partial molar Gibbs energy are identical, that... 

Thus, for the fth component in a system, partial molar internal energy=(dEV ni)jpny Tri = partial molar enthalpy = pH/ ni)rPnin3 = H partial molar entropy = (dS/ dn ... 

Integration of the differential energy of adsorption is quite straightforward from Equation (2.50). Since the gas is ideal, its molar internal energy does not vary with pressure so that ... 

We recall that for condensed phases (either liquid or adsorbed) we are generally able to equate the molar enthalpy and the molar internal energy. Here we shall use enthalpies, simply because they are more common in the literature, but in the following definitions h could be replaced by u. 

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