First Law, Heat Capacity, Latent Heat and Enthalpy

Transcription

1 First Law, Heat Caacity, Latent Heat and Enthaly Stehen R. Addison January 29, 2003 Introduction In this section, we introduce the first law of thermodynamics and examine sign conentions. Heat and Work Heat is the sontaneous flow of energy from one object to another caused by a difference in temerature. Work is defined as any other transfer of energy into or out of the system. Examles of work are: ushing on a iston, stirring a cu of coffee, assing current through a resistor, etc. In each case the temerature will rise as the system s energy is increased. We don t say that the system is being heated because the flow of energy is not sontaneous. Heat and work refer to energy in transit, we can talk about how much energy is in a system, talking about how much work is in a system is meaningless. The First Law of Thermodynamics Let U be the internal energy of a system then U = (Energy inut by Heating) + (Work done on system ). 1

2 This is called the first law of thermodynamics, clearly it is just a statement of the law of conseration of energy. We will use Q, and W to denote large or finite changes, we ll use d Q, and d W to denote infinitesimal changes. Thus we will write U = Q + W, or du = d Q + d W. Where Q, d Q are negatie if heat leaes the system, W, d W are negatie if work is done by the system. Many authors write d Q and d W to denote inexact differentials, and to emhasize that Q or W to take the differential of. That is Q and W are not state functions. When a system changes from state 1 to state 2, while U is determinate, Q and W are ath deendent. Other Sign Conentions There are other sign conentions in use. Many authors define U = Q W so that any work done by the system is ositie. Heat caacity and secific heat In this section we will exlore the relationshis between heat caacities and secific heats and internal. Heat Caacity The heat caacity of an object is the energy transfer by heating er unit temerature change. That is, C = Q T. In this exression, we will frequently ut subscrits on C, C, or C for instance, to denote the conditions under which the heat caacity has been determined. While we will often use heat caacity, heat caacities are similar to mass, that is their alue deends on the material and on how much of it there is. If we are calculating roerties of actual materials we refer 2

3 to use secific heats. Secific heats are similar to density in that they deend only on material. Secific heats are tabulated. When looking them u, be careful that you choose the correct secific heat. As well as tabulating secific heats at constant ressure and constant olume, secific heats are gien as heat caacity er unit mass, heat caacity er mole, or heat caacity er article. In other words, c = C/m, c = C/n, or c = C/N. In elementary hysics mass secific heats are commonly, while in chemistry molar secific heats are common. Be careful! Heat Caacities at Constant Volume and Pressure By combining the first law of thermodynamics with the definition of heat caacity, we can deelo general exressions for heat caacities at constant olume and constant ressure. Writing the first law in the form U = Q V, and inserting this into C = Q T, we arrie at C = U + V. T We can now ealuate this at constant olume, and we arrie at ( ) U C =. T For an infinitesimal rocess we write this as ( ) ( Q ) d Q C = = dt dt or C =. Where as usual the d is used to denote an inexact differential. (By an inexact differential we mean that there is no function to take the differential of, instead the symbol is used to denote a small amount.) Similarly we can calculate the heat caacity at constant ressure. So the heat caacity at constant ressure is gien by ( ) U + V C = T = 3 +.

4 If we examine the exressions for heat caacity, aart from the quantity held constant we see that the exression for C contains an extra term. It isn t too difficult to figure out what is going on. Most materials exand when they are heated. The additional term kees track of the work done on the rest of the unierse as the system exands. Heat Caacities of an Ideal Gas For an ideal gas, we can write the aerage kinetic energy er article as 1 2 m 2 = 3 2 kt. From this, we calculate C and C for N articles. C = = 3 2 Nk To calculate C, we make use of the ideal gas law in the form V = NkT. So, C = + C = 3 2 Nk + (NkT/) = 3 2 Nk + Nk = 5 2 Nk Comaring the exressions for C and C, we see that we can write C C = Nk. So in terms of the ideal gas constant, R, we can write C C = nr. This is an interesting result. It s indeendent of ressure, in other words, at high external ressures, the gas will exand less in such a way that the work done on the enironment is indeendent of. 4

5 Latent heat and enthaly In this section we will deelo the relationshi between latent heat and enthaly. Latent Heat As we hae noted, you can transfer energy by heating without increasing temerature. This haens during hase changes. In a hase change, the heat caacity becomes infinite. The aroriate term to consider is now latent heat. We want to know how much energy is transferred by heating during hase changes. We can define the latent heat as L = Q m = Q N. Some books like to use to use the latent heat er molecule, I hae a reference for using the latent heat er unit mass. The reason isn t dee, latent heats er unit mass are easier to find! The definitions that I hae gien for latent heat are a little ambiguous, in the same way that the definition for heat caacity was ambiguous until I stated the conditions under which it was to be ealuated. By conention, we assume that latent heats are calculated under conditions of constant ressure, and normally that that ressure is one atmoshere. Constant ressure rocesses are common enough that we introduce a new ariable to simlify calculations under constant ressure conditions. That quantity is enthaly. Enthaly Enthaly, H, is defined through H = U + V. It is ossible to use enthaly to urge heat from our ocabulary. I won t do that because most eole still use heat, and you ll need to communicate with others. Basically, the V term in enthaly kees track of exansion/comression related work for us. Starting from the first law in the form du = d Q V 5

6 (Let s call this the first law for hydrostatic rocesses) and the definition C = +, and inserting H, we can rewrite the exression for C in terms of enthaly. Thus, ( ) ( ) (H V) V C = +, or ( ) ( ) ( ) H V V C = +, roducing the result C = + = ( ) H. This exression is often taken as the definition of C. Enthaly and latent heat are simly related. Which, of course is reason that I introduced latent heats in the reious section. The latent heat of aorization L is defined as the difference in enthaly between a fixed mass of aor and the same mass of liquid. That is L = H a H liq. Similarly, the latent heat of fusion is gien by L f = H liq H sol, and the latent heat of sublimations is gien by L s = H a H sol. These enthaly differences are related through L s = L f + L. We shall exlore these relationshis further when we study hase transitions and the Clausius-Claeyron equation later. 6