Thermal & Kinetic Lecture 13

Calculation of entropy,Introduction to 0th law

Recap….

Some abuses of the 2nd law

LECTURE 13 OVERVIEW

Calculation of entropy: reversible process

Section 4: Thermodynamics: the 0th law, equilibrium and isotherms

Last time….

Reversible and quasistatic changes.

The relationship of temperature and the rate of change of entropy wrt energy.

Reversible and irreversible processes: entropy changes.

Reversible and irreversible processes: calculation of entropy

OK, now I’m confused. One of the statements of the 2nd law you’ve given us says that there’s no entropy change for a reversible process. Yet, the expression you’ve written down on the previous slide works ONLY for a reversible process……..?!

373 K

4

293K

1

Reservoir at 373K

293K

2

Reservoir at 373 K

373 K

3

Irreversible process (large temperature differences).

BUT the water at the start and at the end of the process is in equilibrium, with well defined entropies.

Imagine a reversible process that takes the water between the same two end points……………………

Reversible and irreversible processes: calculation of entropy

Reservoir at 293.1 K

293.1 K

Reservoir at 293.2 K

293.2 K

We break the irreversible process down into a series of reversible steps (just as we did when we added weights to the piston).

Reservoir at T+T

T dQ=CPT

Reservoir at 373 K

373 K

If we calculate the entropy change for this process it’s the same as that for the irreversible process.

Reversible and irreversible processes: calculation of entropy

When the water is at a temperature T and it’s heated to T + T, the heat entering (reversibly) is dQ = CPT.

T

dQdS From

the entropy change of the water at each reversible step is: T

dTCdS P

What do we now need to do to evaluate the total change in entropy???

ANS: Integrate. 373

293 T

dTCS P

Calculate the change in the total entropy of the Universe for the following processes:

(i) A block of mass 1 kg, temperature 100°C and heat capacity 100 JK-1 is placed in a lake whose temperature is 10°C.

(ii) The same block at 10°C is dropped into the lake from a height of 10 metres.

(iii) The same block at 10°C absorbs a photon of light ( = 600 nm).

?? Calculating changes in entropy: examples

The change in entropy of the block, SBlock, is given by: 283

373 T

dQSBlock

BlockS 27.61 JK-1

The entropy gain of the lake is:

lakeT

TC

(Lake acts as a thermal reservoir which is so large there’s no change in its temperature).

= 100 x 90/283 = +31.80 JK-1

Calculating changes in entropy: examples

(ii) The block is in the same state (at the same temperature) before and after the process. However, this is obviously an irreversible process. Although the temperature of the lake remains constant because it is a thermal reservoir, the kinetic energy of the block is transferred as heat energy into the lake. So there’s a positive change of entropy for the lake:

lake

lake T

mghS 1 x 9.81 x 10/283 = +0.35 JK-1

Calculating changes in entropy: examples

Some ‘abuses’ of the 2nd law and the concept of entropy

Your sock drawer (or bedroom) does not become disordered due to ‘entropy’ – the change in thermodynamic entropy here is zero (we aren’t changing the number of accessible microstates). (Same thing applies to playing cards!).

“The entropy of a body never decreases – it always increases.”

OK, then how does a fridge work? Heat is ‘taken out’, therefore entropy decreases!

Entropy is a measure of disorder.

Humans and animals are complex, ordered beings.

2nd law states disorder always increases.

Therefore order can’t ‘evolve’ from disorder – theory of evolution can’t be correct…….

Section IV:Thermodynamics from macroscopic

and microscopic perspectives

“….classical thermodynamics has made a deep impression upon me. It is the only physical theory of universal content which I am convinced, within the applicability of its basic concepts, will never be overthrown”. Albert Einstein, 1949

As usual, before starting a new section, let’s recap some of what we’ve covered in previous sections:

S= k ln (W); p = exp (-E/kT);Entropy is a measure of the no. of accessible microstates;At thermal equilibrium entropy is maximised (greatest no. of microstates);Entropy does not change in a reversible process;Temperature is related to the rate of change of entropy with energy;The entropy of a closed system tends to increase.

Introduction to thermodynamics

….but, we still need to address the following questions:

We’ve defined temperature in a couple of different ways, but how do we measure temperature in the real world?

T

dQdS How is an experimental measurement of T related to

the theoretical definition of temperature?

Why can’t we build perpetual motion machines like that shown to the left?

How are heat and work related?

How do heat engines work and what fundamental limits are there on their efficiency?

…and what is Maxwell’s demon?!

Equilibrium and the zeroth law

Thermodynamics was developed before an understanding of the atomic structure of matter was achieved.

Thermodynamics is concerned with the large scale properties of a system: volume, temperature, pressure, specific heat etc…..

Why should we care about thermodynamics? Surely if we can understand the behaviour of matter from an atomic viewpoint using statistical mechanics and quantum theory, that’s enough?!

Thermodynamics is not dependent on a particular microscopic model – as such it acts as an important check on our microscopic description of matter.

For example, the concept of entropy was developed with no consideration of the atomic structure of matter and before the the first law – the conservation of energy – was known!!

First, some revision:

System

Wall may be adiabatic or diathermal

Surroundings

Equilibrium and the zeroth law

Adiabatic wall

E.g. gas isolated from surroundings.

Gas will reach equilibrium state: (i) properties spatially uniform(ii) properties don’t change with time.

Gas

Piston

Equilibrium and the zeroth law

Gas

P, V uniform and constant.

Gas in in the equilibrium state (P,V).

Specifying P, V, and the total number of molecules fixes all the macroscopic properties of the gas (e.g. thermal conductivity).

When thermal equilibrium is reached there are no flows of energy in the system, i.e. no temperature gradients.

To have thermodynamic equilibrium, we must have reached thermal, chemical, and mechanical equilibrium.

Mechanical equilibrium – no unbalanced forces;Chemical equilibrium – number of particles in each phase remains constant (eg. no change in proportion of water and ice as a fn. of time)

Equilibrium and the zeroth law

If two systems are put in thermal contact with each other, they will reach thermal equilibrium (eg blocks considered in earlier lectures on entropy).

0th law of thermodynamics: if each of two systems is in thermal equilibrium with a third, they are in thermal equilibrium with each other.

A

B

C

C

A B

Remember from Section 3: two systems in thermal equilibrium have the same temperature.

TA = TB.

Isotherms

Returning to a consideration of the gas + piston system:

1. Gas is originally in an equilibrium state (P,V) and is in thermal equilibrium with another reference system.

2. Plot point on a graph of P vs V.3. Push piston in to take gas to new equilibrium state (P’, V’) which is also

in equilibrium with the reference system.4. Again, plot this point on a graph of P vs V.5. Repeat 1 – 4, always keeping the temperature constant. The curve is

known as an isotherm.

Isotherms

A functional relationship between P, V and T exists: T = f(P,V) or P = f (T,V) or V = f(P, T)

i.e. of the three measureable variables only two are independent and one may be expressed in terms of the other two.

Functions of state

When a system is in thermodynamic equilibrium the properties of the system only depend on the thermodynamic variables (P, V, T) – the pathway by which equilibrium was reached is irrelevant.

We say that T, V, and P are functions of state.

For example, in equilibrium, T = f (P,V).

Function of stateEquation of state

Write down the equation of state for an ideal gas in the forms P = f(V,T) and T = f(P,V).?? ANS: P = nRT/V, T = PV/nR

Write down an equation which describes the isotherms of an ideal gas. ?? ANS: PV = constant or PV = nRT or

T = PV/nR etc….

Temperature scales

To experimentally measure T first we have to find a physical property of our reference system that varies with temperature: e.g. length of a column of mercury in a glass capillary, voltage of a thermocouple junction.

Second, we need a temperature scale based on the change in the physical property. How do we assign a scale?

Let’s assume that the physical property (X) varies linearly with temperature (T) so that:

X = c TX

The value of c is fixed by choosing a certain well defined and reproducible TX and assigning it a certain value.

Temperature scales

Now, which point do we choose as our fixed point?

We choose the point at which ice, water, and water vapour coexist in equilibrium – the triple point of water. (More on phases later…….)

Then we assign TX at the triple point of water the value 273.16 K (i.e. 0.01°C)

Ermmm…….?! Yes, I know – we’ll explain that 273.16 K figure soon. For now, take on board the following:

(i) X = cTx implies that as Tx 0, X 0 – not necessarily. (ii) Temperatures on the X scale are only defined in certain regions.(iii)Different thermometers based on different variables will agree only

at fixed points (deviations from linearity), except for…….