JIF 314Thermodynamics

Chapter 4

Heat and the first law of thermodynamics

Key concepts

Work and heat Adiabatic work Work done adiabatically is path

independence Internal energy function, U is a state fuction

of the system. Wif (adiabatic) = Uf – Ui Definition: Uf >Ui when work is done on the

system.

Internal energy function

Interpretation of U = Uf – Ui Conservation of energy Difference in energy function = energy

change in the system U a function of two thermodynamical

coordinates, e.g. {P,T}, {P,V},{V,T} The third variable is fixed by the equation of

state.

U is specified by any two thermodynamical coordinates (out of three) Example: Given the equation of state, PV = RT, there are two

degree of freedom with three variables. Out of three variable, only two are independent since the equation of state fixes the third one.

For example, if we choose {V ,T } as the two independent variables, P is then the dependent variable that is fixed by the equation of state via

P =RT / V.

Alternatively, we can also choose {P ,T } as the two independent variables, V is then the dependent variable via the equation of state,V = RT /P .

As a conclusion, to specify the state of U, we need only any pair of independent thermodynamical coordinates

U is path-independent

U is path independent, hence dU is an exact differential Say U (a, b), U (a+da, b+db), a,b any two thermodynamical

coordinates dU = U (a+da, b+db) - U (a, b)

fixed a fixed

d d db

U UU a b

a b

{a, b} can be e.g. {T, V } or {T, P } or {V, P }. In each case, the third variable, c, are P, V and T

respectively.

fixedb

U

a

fixedb

U

a

are two different functions

U = U (a,b) U = U (a,c )

fixedb

U

a

fixedc

U

a

≠

fixedb

U

a

fixed

, c

U

a

Non-adiabatical process

Diathermal wall For non-adiabatic process, Wif ≠ Uf – Ui

The difference between Wif and Uf – Ui is called heat, Q = (Uf – Ui) - Wif

Convention: Q is positive if heat enters a system, negative when if leave the system

Transit of heat is a result of a temperature difference Heat is a form of energy

U i

Q

U i U f

Volume expand from Vi to Vf , hence work is done by the system. Internal energy increases from Ui to Uf when positive Q flows in. Work is done by the system.

weight

Volume expands

Q = (Uf – Ui) - W

Temperature Tsu > Tsy so that heat flows from surrounding (su) to the system (sy)

Diathermal wall permitting heat flow

Q and W have meaning only if a state undergoes transitional process to a new one Heating and working are transient processes that

causes a system to change from one state to another.

Heat and work are involved only in the process of making transition from a state to another.

Once the transition of states ceases and equilibrium achieved, heat or work exist no more.

Once the transition of state ceases, what endures finally is the new state, and the final internal energy.

Infinitesimal amount of Q, W are not exact differentials U U is a state function of the coordinates of the is a state function of the coordinates of the

system, hence, it is path-independentsystem, hence, it is path-independent The difference in The difference in U U between two between two

infinitesimally different states is an exact infinitesimally different states is an exact differential, ddifferential, dUU, and we can write, e.g., and we can write, e.g.

dPP

UdT

T

UdU

TP

Infinitesimal amount of Q, W are not exact differentials In contrast, Q and W are not state function,

path-dependent. The difference in Q and W between two

infinitesimally different states are not exact differential,

that is, e.g., dPP

QdT

T

QdQ

TP

Inexact differential form of Q and W

Hence, we use to denote an infinitesimal amount of heat, and not the differential form, dQ.

The same notation goes to W.

Qd

Calculation of W and Q are path-dependence What all these mean are that: the calculation

involving heat and work is path-dependent, and normally we have to carry out integration to determine W and Q. between two states, that is path-dependent.

Path-independence and path-dependence

As an example, when we calculate the difference in internal energy between two states, we only need to calculate the difference, U =Uf – Ui. This difference is always the same since U is a state function. This infers path-independence.

However, in calculating the work done, W when a system change from one state to another, we cannot simply calculate W as Wf – Wi but we have to perform the integration which will result in different value for process carried out via different route (e.g. adiabatical path result in a final work done that is different from that is a non-adibatical one)

path 1 path 2dW dW

Within an adiabatic boundary, the heat lost (or gained) by system A is equal to the heat gained (or lost) by system B

System A

System B

Q = - Q’

Q, heat flom into system A

diathermal wall

Adiabatic wall

Q’, heat flow into system B

Differential form of the first law Two inexact differentials on the right hand

side (RHS) make one exact differential on the LHS.

For hydrostatic system (fluid),

and the first law reduces to

dU dQ dW

ddW P V

dU dQ PdV

Work done is path dependent

Heat Capacity

In unit of joules per kelvin (J/K) It is a extensive quantity (i.e. the larger the

mass the larger is the value of C since a larger amount of heat is require to heat up the material for 1 degree.)

limi f

i

T Tf i T

Q dQC

T T dT

Specific heat Capacity

In unit of joules per kelvin per kg (J/kg∙K) Intensive quantity, i.e. it’s value remains the

same for different amount of mass of the same material.

/c C m

Molar heat capacity

n is the amount of material measured in unit of mole. In unit of joules per kelvin per mole (J/mol∙K) Intensive quantity, i.e. it’s value remains the same

for different amount of mass of the same material.

/c C n

Amount of material in mole

1 mole of material = NA atom NA = Avogardo number, 6.023 1023

If an atom has a mass of m, N atoms will have a total mass of M = mN

Given an element with atomic mass m, M kg of material made up of such element contains

N = M/m atoms. The ratio of N/NA defines the amount of atom

in mole of that material: n = N/NA

Heat capacity at constant pressure

P P ,P

dQC C P T

dT

Heat capacity at constant volume

V V ,V

dQC C V P

dT

Equations for a hydrostatic system

If we choose U=U(T,V)

dU dQ PdV

V T

V T

V T

V T

U UdU dT dV

T V

U UdQ PdV dT dV

T V

U UdQ dT P dV

T V

dT

dQ U U dVP

dT T V dT

Special case, dV = 0 (for the case of CV) Raising the temperature by heating

substance but without changing the volume:

VV T VV

dQ U U dV dQ UP C

dT T V dT dT T

Specific heat at constant volume of a substances CV can

be calculated from theory if the internal energy function of that substance is known.

Special case, dP = 0 (for the case of CP) Raising the temperature by heating

substance but without changing the pressure:

V T

PV T PP

P VT

P V

T

dQ U U dVP

dT T V dT

dQ U U VC P

dT T V T

UC C P V

V

C C PVU

V V

Specific heat at constant pressure of a substances CP can be calculated from theory if the internal energy function and of that substance is known.

Heat reservior

A body of such a large mass that it may absorbed or reject an unlimited quantity of heat without experiencing an appreciable change in temperature or in any other thermodynamic coordinate.

Calculating quasi-static isobaric heat transfer process via a temperature difference

If CP is constant in temperature in the range of Ti – Tf,

f

i

T

P PTQ C dT

P P f iQ C T T

Calculating quasi-static isochoric heat transfer process via a temperature difference

If CV is constant in temperature in the range of Ti – Tf,

f

i

T

V VTQ C dT

V V f iQ C T T

Three mechanism of heat conduction Conduction Convection Radiation

Heat conduction

dQ dTKA

dt dx

Thermal conductivity

Temperature gradient

Heat flow from high temperature to low temperature

Cross section perpendicular to direction of heat flow

Heat convection

dQhA T

dt

Convection coefficient

Temperature difference

Thermal radiation Emission of heat as electromagnetic radiation Absorbitivity Radiant exitance, R Emissivity, Black body Kirchhoff’s law Radiated heat

bb W bb

dQA T R T R T

dt

Stefan-Boltzmann law

4bbR T TStefan-Boltzmann constant, = 5.67051 10-8 W/m2∙K4

4 4W

dQA T T

dt

Experimental determination of Nonequilibrium method Equilibrium method

Question 4.1 Regarding the internal energy of a hydrostatic system to be a function of T and P, derive the following equations: a) b) c)

Solution for 4(a)

U =U (T, P ) First law of Thermodynamics => Combining both Eq. (1)

Solution for 4(a) (cont.)

For a hydrostatic system or PVT system, we can write V as a function of T and P.

By substituting the expression of dV into equation Eq(1), we get

Eq. (2)

Solution for 4(b)

At constant pressure, dP=0. Setting dP=0, and dividing Eq. (2) by dT, we get

Since

Therefore,

PP

dQC

dT

Solution for 4(c) At constant volume, dV=0. Setting dV=0, and dividing

Eq. (1) by dT, we get

V V

dP P

dT T

VV

;P T V P T V

dQ U U dP U U PC

dT T P dT T P T

1 1

P P P

V T TV V

V T V V

Eq. (3)

Eq. (4)

Solution for 4(c) Combining Eq. (3), (4), and

VV

PP

T T

C C PVU UC C PV

P P

Question 4.3

One mole of a gas obeys the van der Waals equation of state:

and its molar internal energy is given by where a,b,c and R are constants. Calculate the molar heat capacities cv and cP.

Solutions

We write U = U(T,V)

v T

u udu dT dv

T v

Eq. (1)

Eq. (2)

Eq. (1) combined with Eq. (2)

v T

u udq dT P dv

T v

Vv

uc

T

dTv T

dq u u dvP

dT T v dT

Eq. (5)

dq du Pdv

Solutions At constant volume, Eq. (5) becomes

,since /Vv v

dq uc c u cT a v

dT T

Solutions At constant volume, Eq. (5) becomes

2since /

PP V T P

P V VT P P

dq u u dvc P

dT T v dT

u v a vc c P c P u cT a v

v T v T

From

2

2

2 3

2P VP

aPa V VC C P Ra v bV T a

Pv v

2 3

2P

v Ra v bT a

Pv v