Internal Energy in terms of Properties

Transcription

1 Lecture #3 Internal Energy in terms of roerties Internal energy is a state function. A change in the state of the system causes a change in its roerties. So, we exress the change in internal energy in terms of the changes in its roerties. Internal energy is a function of temerature and volume of the system. E = E(, ) de = ( ) d + ( ) d his equation indicates that if the temerature of the system increases by an amount d and the volume increases by an amount d then the total increase in energy is the sum of two contributions. ( ) ( ) : rate of increase in E with temerature at constant volume : rate of increase in E with volume at constant temerature In order to calculate de we must exress these artial derivatives in terms of measurable quantities. onstant olume rocesses By using the first law equation, it may be written; dq ' d = ( ) d + ( ) d

2 At constant volume d=0 dq E ( ) = ( ) = dq d v d Heat absorbed er degree rise in temerature is the constant volume heat caacity. is defined as the change in internal energy with temerature at constant volume. = ( ) Using the definition of we can write; de = d or integrating we have E = d Internal energy is an extensive roerty, so is also an extensive roerty. Generally, molar heat caacity is used. If is constant in the temerature range investigated, the change in internal energy can be calculated as; E = Joule s Rule he identification of the second artial derivative is not so easily managed. For

3 A B = gases it can be done by an exeriment devised by Joule. In this exeriment, two containers are connected through a stocock. In the initial state, A is filled with a gas while B is evacuated. he aaratus is immersed in a water bath and is allowed to equilibrate with water at. he water is continuously stirred. he initial temerature is read on the thermometer. hen, the stocock is oened and the gas exands to fill B. After allowing time for the system to come to thermal equilibrium, the temerature is read again. It was observed that there is no temerature change. In this exeriment, no work is roduced in the surroundings. Since B is evacuated = 0. here is no oosing force against the exanding gas. his is called free-exansion. First law equation takes the form; de = dq Since the temerature does not change dq = 0, then de = 0. de = ( ) d = 0 In the exeriment d 0, so; ( ) = 0 he energy is indeendent of the volume. Joule s rule states that the internal energy of an ideal gas is a function only of temerature. Joule s rule is not correct for real gases. For real gases, it is greater than zero. For liquids and solids, the volume change is very small. Even though the value of the 3

4 E = d derivative is very large, the value of is so small that the roduct is very nearly zero. herefore, the internal energy of all substances is considered to be a function only of. onstant ressure rocesses Most of the exeriments are carried out in the laboratory at constant ressure. At equilibrium, the external ressure is equal to the ressure of the system. For constant ressure rocesses, the first law equation may be written as; de = dq d Since is constant, the integration gives; de = dq d E E = Q ( ) Rearranging, we obtain; ( E + ) ( E + ) = Q and are roerties of the system. E is a state function. So, the function (E + ) is itself a state function. It deends uon the state of the system. his function is called enthaly and designated by H. Enthaly is defined as; H = E + Enthaly is an extensive roerty. By using the definition of enthaly, we can rewrite Q as; H H = Q or H = Q 4

5 his equation indicates that in constant ressure rocesses heat absorbed by the system is equal to the increase in its enthaly. For an infinitesimal change; dh = dq Since H is a state function, dh is an exact differential. H is a function of and. otal differential of H may be written as; dh = d + d For a constant ressure rocess d = 0 and dh = dq ; dq = d dq /d is the amount of heat absorbed er degree rise in temerature. It is the constant ressure heat caacity. So, the definition of may be written as; = herefore the total differential of H takes the form; dh = d + d For constant ressure rocesses; dh = d H = d 5

6 If is constant in the temerature range; H = Enthaly hange with ressure Definition of H is; H = E + Differentiating we obtain; dh = de + d + d We can write dh and de in terms of the roerties of the system; dh = d + d de = d + d Substituting these equations in dh equation we obtain; d + d = d + d + d + d We are calculating the change in enthaly with ressure at constant temerature. So, d = 0. d = + d + d Dividing both sides by d we obtain; 6

7 d = + + Since solids and liquids are incomressible hases, their volume does not change much with ressure. herefore, the first term in the above exression is negligible. = Moreover, molar volume of solids and liquids is very small. If the ressure change is not too large, the change in enthaly with ressure can be ignored. = R For ideal gases; Substituting this equation into the definition of enthaly we obtain; H = E + = E + R Since the internal energy is a function only of temerature, H for an ideal gas is also a function only of temerature. For a real gas, enthaly is a function of both temerature and ressure. Joule-homson Exeriment For a real gas, ( H / ) can be measured by Joule-homson exeriment. In this exeriment, an insulated ie is used. here is an obstruction in the middle of the ie with a hole on it. Gas flows in the direction of the arrows. Gauges and thermometers are laced in the ie to measure the changes in and. 7

8 Due to the resence of the obstruction, the ressure of the gas decreases in assing from left to right. If one mole of gas asses from left to right, the gas in the left-hand side roduces work equal to while the gas in the right-hand side gains energy equal to. he mole of gas is ushed by the gas behind in the left side, while it ushes the gas ahead in the right side. Since the ie is insulated Q = 0. W = + E = E E = E + + = E H = H In Joule-homson exeriment, enthaly remains constant, so dh = 0. 0 = d + d d d = H he limit of this ratio is known as Joule-homson oefficient and designated by µ. 8

9 µ J = H Substituting into the above equation we obtain; = µ J µ changes sign at Joule-homson inversion temerature. Above this temerature J µ < 0, below it is ositive. For all gases excet H and He, J is above room J temerature. Gas ressure and gas temerature change in the same direction. So, if the ressure of the gas is decreased, its temerature also decreases. 9