Dr Roger Bennett Rm. 23 Xtn Lecture 13

Entropy changes – the universe Imagine heating a beaker of water by putting it in contact with a reservoir whose temperature remains fixed and the system remains at constant pressure. In lecture 11 we calculated the entropy change under different process conditions. However as entropy is a function of state the change in entropy is defined by the end points not the process. What about the reservoir itself?

Entropy changes The reservoir donates heat Q = C p (T f - T i ) at the temperature of the reservoir T f. The total entropy change of the universe (system + reservoir) can therefore be calculated. Plugging in T i = 273K and T f = 373K

Entropy changes The entropy of the universe increases as we expect. What happens if we employ an intermediate reservoir to get the water to 50°C and then the original reservoir to 100 °C? Plugging in T i = 273K, T 1 = 323K and T f = 373K

Entropy changes The entropy of the universe increases as we expect but by using two reservoirs the entropy change is reduced. Why? What happens if we have an infinite number of reservoirs that track the temperature of the water? This was our original version in lecture 11 – we used it to calculate the reversible change in entropy. The total change of the entropy of the universe for the reversible process is zero.

Enthalpy Many experiments are carried out at constant pressure –worthwhile developing a method. Changes in volume at constant pressure lead to work being done on or by the system. –dU = đQ R - PdV The heat added to the system at constant pressure can be written. –đQ R = dU + PdV = d(U + PV) which is true as undertaken at constant pressure so dP = 0. H = U + PV, is called the Enthalpy –dH = d(U + PV) = dU + VdP + PdV = dQ R + VdP –This is a general expression, the enthalpy is a state function with units of energy.

Enthalpy Change in volume at constant pressure – changes of phase, boiling, melting or changing crystal structure. E.g. 1 mole of CaCO 3 could change crystal structure at 1 bar (10 5 Pa) with an increase in internal energy of 210 J and a density change from kgm -3. The change in enthalpy is:- H = (34-37) = J. H is very similar to U as the PV term is small at low pressure for solids. This isnt true at high pressure or for gases.

Measuring Entropy and Latent Heats Consider heat being added reversibly to a system which is undergoing a phase transition. At the transition temperature heat flows but the temperature remains constant. The H that flows is the latent heat. +values are endothermic –ve values exothermic.

Measuring Entropy To measure the entropy assume the lowest temp. we can attain is T 0 and we measure C p as a function of T from T 0 up to T f.

Measuring Entropy Cp(T) can typically be measured by calorimetry except at absolute zero as we cannot get there. Most non-metallic solids show C p (T) = aT 3 so we can extrapolate to find S(0). These tend to converge on the same value of S(T=0) so we define S(T=0).

Third Law of Thermodynamics If the entropy of every element in its stable state at T=0 is taken to be zero, then every substance has a positive entropy which at T=0 may become zero, and does become zero for all perfectly ordered states of condensed matter. Such a definition makes the entropy proportional to the size (number elements) in the system – ie entropy is extensive. By defining S(0) we can find entropy by experiment and define new state functions such as the Helmholtz free energy F = U-TS. This is central to statistical mechanics that we will develop in detail.

Dr Roger Bennett Rm. 23 Xtn Lecture 14

Third Law of Thermodynamics If the entropy of every element in its stable state at T=0 is taken to be zero, then every substance has a positive entropy which at T=0 may become zero, and does become zero for all perfectly ordered states of condensed matter. Such a definition makes the entropy proportional to the size (number elements) in the system – ie entropy is extensive. By defining S(0) we can find entropy by experiment and define new state functions such as the Helmholtz free energy F = U-TS. This is central to statistical mechanics that we will develop in detail.

Properties of Entropy Entropy is a state function. Entropy increases during irreversible processes. Entropy is unchanged during reversible processes. E.g. reversible adiabatic expansion. Constant Entropy processes are isentropic. Entropy is extensive. UextensiveConstraint VextensiveConstraint nextensiveConstraint SextensiveEquilibrium PintensiveEquilibrium TintensiveEquilibrium

Expansion of an ideal gas - microscopic Expansion of ideal gas contained in volume V. U, T unchanged and no work is done nor heat flows. Entropy increases – what is the physical basis? Gas in V Gas in 2V

Free expansion No temperature change means no change in kinetic energy distribution. The only physical difference is that the atoms have more space in which to move. We may imagine that there are more ways in which the atoms may be arranged in the larger volume. Statistical mechanics takes this viewpoint and analyses how many different states are possible that give rise to the same macroscopic properties.

Statistical View The constraints on the system (U, V and n) define the macroscopic state of the system (macrostate). We need to know how many microscopic states (microstates or quantum states) satisfy the macrostate. A microstate for a system is one for which everything that can in principle be known is known. The number of microstates that give rise to a macrostate is called the thermodynamic probability,, of that macrostate. (alternatively the Statistical Weight W) The largest thermodynamic probability dominates. E.g. think about rolling two dice, how many ways are there of rolling 7, 2 or 12? The essential assumption of statistical mechanics is that each microstate is equally likely.

Statistical View Boltzmanns Hypothesis: The Entropy is a function of the statistical weight or thermodynamic probability: S = ø(W) If we have two systems A and B each with entropy S A and S B respectively. Then we expect the total entropy of the two systems to be S AB = S A + S B (extensive). Think about the probabilities. W AB = W A W B So S AB = ø(W A ) + ø(W B ) = ø(W AB ) = ø(W A W B )

Statistical View Boltzmanns Hypothesis: S AB = ø(W A ) + ø(W B ) = ø(W AB ) = ø(W A W B ) The only functions that behave like this are logarithms. S = k ln(W) Boltzmann relation The microscopic viewpoint thus interprets the increase in entropy for an isolated system as a consequence of the natural tendency of the system to move from a less probable to a more probable state.

Expansion of an ideal gas - microscopic Expansion of ideal gas contained in volume V. U, T unchanged and no work is done nor heat flows. Entropy increases – what is the physical basis?

Expansion of an ideal gas - microscopic Split volume into elemental cells V. Number of ways of placing one atom in volume is V/ V. Number of ways of placing n atoms is –W = (V/ V) n S = nk ln(V/ V) –Is this right? Depends on the size of V.

Expansion of an ideal gas - microscopic Is this right? Depends on the size of V. Yep, we only measure changes in entropy. S f -S i =nk(ln (V f / V) - ln (V i / V))= nk ln(V f /V i ) Doubling volume gives S = nk ln(2) = NR ln(2)