Energy of an ideal gas

Said another way, the internal energy for an ideal gas is independent of the position of the molecules. In Chapter 4, we will consider the thermodynamic properties of real 

In Example 2.1, we considered the potential energy of a stone at the top of a 10-m chff. When it fell, it gained kinetic energy, resulting in a velocity around 31 miles/hr. Consider now an equivalent mass of water initially at 25 C. How hot would the water end up if its internal energy increased by the same amount  

SOLUTION If we write Equation (2.2) on a per-mass basis, we have  

We have used specific energy and converted the units to be consistent with the steam tables. Again we will use state 1 to denote the initial state and state 2 to denote the final state. liquid water is subcooled at 25 C and 1 atm however, we do not expect the properties of a fiquid to be significantly affected by pressure. Therefore, we can use the temperature tables for saturated liquid water at 25 XZ (which is technically at a pressure of 0.03 atm). From Appendix B.l  

The problem statement says the internal energy of the water increases by the same amount as the energy of the stone, that is. 

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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... 

The entropy and Gibbs energy of an ideal gas do depend on pressure. By equation 85 (constant T),... 

For the Gibbs energy of an ideal gas mixture, — T the parallel relation for partial properties is equation 149 ... 

It therefore follows that /ree energy of an ideal gas mixture is a function of the same form as that of a simple gas. Hence, in virtue of Massieu s theorem, an ideal gas mixture behaves thermally and mechanically exactlylike a simpler gas. 

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 Gibbs free energy of a liquid is almost independent of pressure, and so we can replace Gm(l) by its standard value (its value at 1 bar), Gm°(l). The Gibbs free energy of an ideal gas does vary with pressure, and thermodynamics can be used to show that, for an ideal gas,... 

FIGURE 8.4 The variation of the molar Gibbs free energy of an ideal gas with pressure. The Gibbs free energy has its standard value when the pressure of the gas is 1 bar. The value of the Gibbs free energy approaches minus infinity as the pressure falls to zero. 

To find how AG changes with composition, we need to know how the molar Gibbs free energy of each substance varies with its partial pressure, if it is a gas, or with its concentration, if it is a solute. We have already seen (in Section 8.3) that the molar Gibbs free energy of an ideal gas J is related to its partial pressure, P(, by... 

The spectroscopic dissociation energy D is the dissociation energy of an ideal gas molecule at absolute zero, where all the gas molecules are in the zero potential energy level, h is Planck s constant (6.62 x lO Vrg second), and iv, is the frequency of vibration of the nuclei at the lowest vibrational level, which is above the point of zero potential energy at the equilibrium intemuclear separation. Thus, for the hydrogen molecule. D — 4.476 electron volts, l o = 1.3185 x IO 1 sec1, and since I electron volt = 23.06 kilocalories per mole wc calculate I) using Hq.(21) as... 

Setting the kinetic energy equal to the Coulomb barrier as would be appropriate when all the particles are moving (in a gas), and using the thermal energy of an ideal gas ... 

We have used M = mNA in the last step. The molar kinetic energy of an ideal gas is the mean energy of a molecule (the quantity we have just calculated) multiplied by the number of molecules per mole (the Avogadro constant). Therefore,... 

We prove the identity of the Kelvin scale and the ideal gas scale by using an ideal gas as the fluid in a reversible heat engine operating in a Carnot cycle between the temperatures T2 and 7. An ideal gas has been defined by Equations (2.36) and (2.37). Then the energy of an ideal gas depends upon the temperature alone, and is independent of the volume. 

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. 

Where G° is molar free energy of an ideal gas in its standard state and G is molar free energy of an ideal gas in the state of interest. 

We consider the sum of states, density of states, and energies of an ideal gas in a box of volume V. The Hamiltonian for a free particle of mass m is... 

Using Eq. (5) of Chapter 2 and the fact that the energy of an ideal gas depends only on its temperature,... 

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. 

Recall that the average translational energy of an ideal gas E is given by the... 

In this section we will lay the groundwork for several fundamental concepts of thermodynamics by considering the isothermal expansion and compression of an ideal gas. An isothermal process is one in which the temperatures of the system and the surroundings remain constant at all times. Recall that the energy of an ideal gas can be changed only by changing its temperature. Therefore, for any isothermal process involving an ideal gas,... 

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