03. First Law, Heat Capacity and Enthalpy

First Law of Thermodynamics,Heat Capacity, Latent Heat and Enthalpy

Stephen R. Addison

University of Central Arkansas

January 17, 2013

Outline

1 IntroductionHeat and Work

2 The First Law of ThermodynamicsSign Conventions

3 Heat capacity and specific heatWater and CalorimetryHeat Capacities at Constant Volume and PressureHeat Capacities of an Ideal GasRatio of Specific Heats

4 Latent heat and enthalpy

Stephen R. Addison (UCA) First Law of Thermodynamics January 17, 2013 2 / 30

Introduction

Introduction

In this lecture, we introduce the first law of thermodynamics and examinesign conventions.


Introduction Heat and Work

Heat and Work

Heat is the spontaneous flow of energy from one object to another causedby a difference in temperature. Work is defined as any other transfer ofenergy into or out of the system. Examples of work are: pushing on apiston, stirring a cup of coffee, passing current through a resistor, etc. Ineach case the temperature will rise as the system’s energy is increased. Wedon’t say that the system is being heated because the flow of energy is notspontaneous.Heat and work refer to energy in transit. While we can talk about howmuch energy is in a system, talking about how much work is in a system ismeaningless.


The First Law of Thermodynamics


Let E be the internal energy of a system then

4E = (Energy input by Heating) + (Work done on system ).

This is called the first law of thermodynamics, clearly it is just arestatement of the law of conservation of energy. We will use Q, and Wto denote large or finite changes.



Inexact Differentials

We’ll use d−Q, and d−W to denote infinitesimal changes. Thus we will write

4E = Q + W ,

ordE = d−Q + d−W .

Where Q, d−Q are negative if heat leaves the system, W , d−W are negativeif work is done by the system. Many authors write d−Q and d−W to denoteinexact differentials, and to emphasize that there is no function Q or W totake the differential of. That is Q and W are not state functions.



When a system changes from state 1 to state 2, while 4E is determinate,Q and W are path dependent.


The First Law of Thermodynamics Sign Conventions

Sign Conventions

There are other sign conventions in use. Many authors define4E = Q −W so that any work done by the system is positive. My adviceon this sign convention is the same as my advice on all other signconventions – you should know which sign convention you are using andyou should use it consistently.


Heat capacity and specific heat

Heat Capacity and Specific Heat

In this section we will explore the relationships between heat capacitiesand specific heats, and internal energy.


Heat capacity and specific heat Water and Calorimetry

Water and Calorimetry

The amount of energy required to raise the temperature of one gram ofpure liquid water from 14.5oC to 15.5oC is one calorie. This is equivalentto defining the heat capacity of one gram of water under these conditionsto be one calorie. In general the heat capacity of any object is the ratio ofheat absorbed to temperature increase.



Heat Capacity

The heat capacity of an object is the energy transfer by heating per unittemperature change. That is,

C =Q

4T.

In this expression, we will frequently put subscripts on C , Cp, or Cv forinstance, to denote the conditions under which the heat capacity has beendetermined. While we will often use heat capacity, heat capacities aresimilar to mass, that is their value depends on the material and on howmuch of it there is.



Specific Heat

If we are calculating properties of actual materials we prefer to use specificheats. Specific heats are similar to density in that they depend only onmaterial. Specific heats are tabulated. When looking them up, be carefulthat you choose the correct specific heat. As well as tabulating specificheats at constant pressure and constant volume, specific heats are given asheat capacity per unit mass, heat capacity per mole, or heat capacity perparticle. In other words, c = C/m, c = C/n, or c = C/N. In elementaryphysics mass specific heats are commonly, while in chemistry molarspecific heats are common. Be careful!



So using differentials, we could write:

C =d−Q

dT,

cm =1

m

(d−Q

dT

),

or

cmol =1

n

(d−Q

dT

).

Where, as usual, the “d−” is used to denote an “inexact” differential. (Byan inexact differential we mean that there is no function to take thedifferential of, instead the symbol is used to denote a small amount.)


Heat capacity and specific heat Heat Capacities at Constant Volume and Pressure

Heat Capacities at Constant Volume and Pressure

By combining the first law of thermodynamics with the definition of heatcapacity, we can develop general expressions for heat capacities atconstant volume and constant pressure. If we apply a pressure p to apiston in a gas filled cylinder, the gas will be compressed, the work on onthe gas is then W = −p4V . So Writing the first law in the form4E = Q − p4V , and inserting this into C = Q

4T , we arrive at

C =4E + p4V

4T.



We can now evaluate this at constant volume, and we arrive at

Cv =

(4E

4T

)v

.

For an infinitesimal process we write this as

Cv =

(d−Q

dT

)v

or

Cv =

(∂E

∂T

)v

.



Similarly we can calculate the heat capacity at constant pressure. So theheat capacity at constant pressure is given by

Cp =

(4E + p4V

4T

)p

=

(∂E

∂T

)p

+ p

(∂V

∂T

)p

.



If we examine the expressions for heat capacity, apart from the quantityheld constant, we see that the expression for Cp contains an extra term. Itisn’t too difficult to figure out what is going on. Most materials expandwhen they are heated. The additional term keeps track of the work doneon the rest of the universe as the system expands.


Heat capacity and specific heat Heat Capacities of an Ideal Gas

Heat Capacities of an Ideal Gas

For an ideal gas, we can write the average kinetic energy per particle as

1

2m〈v 2〉 =

3

2kT .

This comes from combining our calculation that

p =1

3ρ〈v 2〉

with

ρ =M

V=

Nm

Vand pV = NkT

so that

pV = NkT =1

3Nm〈v 2〉 or 3kT = m〈v 2〉,

and the result follows. (Note, as promised I have shown that E = E (T )for an ideal gas.)



Our result is true for a monatomic gas, for a diatomic gas the internalenergy is 5

2 NkT and for a polyatomic gas it is 62 NkT - this is because the

individual particles have more degrees of freedom and the average energyper molecule is kT

2 per degree of freedom. We have to consider rotationalkinetic energy in diatomic and polyatomic molecules.



We now calculate Cv and Cp for N particles.

Cv =

(∂E

∂T

)v

=3

2Nk

To calculate Cp, we make use of the ideal gas law in the form pV = NkT .

Cp =

(∂E

∂T

)p

+ p

(∂V

∂T

)p

So,

Cp =3

2Nk + p

∂

∂T(NkT/p)p =

3

2Nk + Nk =

5

2Nk



Comparing the expressions for Cp and Cv , we see that we can write

Cp − Cv = Nk .

So in terms of the ideal gas constant, R, we can write

Cp − Cv = nR.

This is an interesting result.



We should note that Cp = 72 R in diatomic gases and Cp = 8

2 R. (Therelationship between Cp and Cv still holds in those cases.)


Heat capacity and specific heat Ratio of Specific Heats

The ratio of specific heats

The ratio of heat capacities at constant volume and pressure is equal to theratio of specific heats under the same conditions as the amount of materialis eliminated when we form the ratio. Thus we can write the ratio as

γ =Cp

Cv=

cpcv.

The values of γ are then 5/3 (monatomic gas), 7/5 (diatomic gas) and4/3 (polyatomic gas).


Latent heat and enthalpy

Latent Heat and Enthalpy

We now explore the the relationship between latent heat and enthalpy.



Latent Heat

It is possible to transfer energy by heating without increasing temperatureof a material. This happens during phase changes. In a phase change, theheat capacity becomes infinite. The appropriate term to consider is nowlatent heat. We want to know how much energy is transferred by heatingduring phase changes.



We can define the latent heat as

L =Q

m=

Q

N.

Some books like to use to use the latent heat per molecule, I prefer to usethe latent heat per unit mass. The reason isn’t deep, latent heats per unitmass are easier to find! The definitions that I have given for latent heatare ambiguous. (In the same way that the definition for heat capacity wasambiguous). By convention, we assume that latent heats are calculatedunder conditions of constant pressure, and normally that that pressure isone atmosphere. Constant pressure processes are common enough that weintroduce a new variable to simplify calculations under constant pressureconditions. That quantity is enthalpy.



Enthalpy

Enthalpy, H, is defined through

H = E + pV .

It is possible to use enthalpy to purge heat from our vocabulary. I won’tdo that, because most people still use heat, and you’ll need tocommunicate with others. Basically, the pV term in enthalpy keeps trackof expansion/compression related work for us.



Starting from the first law in the form

dE = d−Q − p4V

(Let’s call this the first law for hydrostatic processes) and the definition

Cp =

(∂E

∂T

)p

+ p

(∂V

∂T

)p

,

and inserting H, we can rewrite the expression for Cp in terms of enthalpy.



Thus,

Cp =

(∂(H − pV )

∂T

)p

+ p

(∂V

∂T

)p

,

or

Cp =

(∂H

∂T

)p

− p

(∂V

∂T

)p

+ p

(∂V

∂T

)p

,

producing the result

Cp =

(∂E

∂T

)p

+ p

(∂V

∂T

)p

=

(∂H

∂T

)p

.

This expression is often taken as the definition of Cp.



Enthalpy and latent heat are simply related. The latent heat ofvaporization Lv is defined as the difference in enthalpy between a fixedmass of vapor and the same mass of liquid. That is Lv = Hvap − Hliq.

Similarly, the latent heat of fusion is given by Lf = Hliq − Hsol , and thelatent heat of sublimations is given by Ls = Hvap − Hsol . These enthalpydifferences are related through Ls = Lf + Lv . We shall explore theserelationships further when we study phase transitions and theClausius-Clapeyron equation later.


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03. First Law, Heat Capacity and Enthalpy