Short Version : 18. Heat, Work, & First Law of Thermodynamics

18.1. The 1st Law of Thermodynamics

Either heating or stirring can raise T of the water.

Joule’s apparatus

1st Law of Thermodynamics:Increase in internal energy = Heat added Work done

U Q W

Thermodynamic state variable = variable independent of history.

e.g., U, T, P, V, …

Not Q, W, …

dU dQ dW

dt dt dt

PE of falling weight

KE of paddle

Heat in water

18.2. Thermodynamic Processes

Quasi-static process: Arbitrarily slow process such that system always stays arbitrarily close to thermodynamic equilibrium.

Reversible process:Any changes induced by the process in the universe (system + environment) can be removed by retracing its path.

Reversible processes must be quasi-static.

Irreversible process:Part or whole of process is not reversible.

e.g., any processes involving friction, free expansion of gas ….

Twater = Tgas & rises slowly

system always in thermodynamic equilibrium

Work & Volume Changes

W F x p A x p V

W d W2

1

V

Vp dV

面積

Work done by gas on piston

extW F x

Isothermal Processes

Isothermal process: T = constant.

2

1

V

VW p dV 2

1

V

V

n R TdV

V 2

1ln V

Vn R T V

2

1

lnV

W n R TV

3

2U N k T 0U Q W

2

1

lnV

Q W n R TV

Isothermal processes on ideal gas

Constant-Volume Processes & Specific Heat

Constant-volume process ( isometric, isochoric, isovolumic ) : V = constant

0V 0W p V

U Q

VU Q n C T

CV = molar specific heat at constant volume

Ideal gas: U = U(T) ideal gas VU n C T for all processes

isometric processes

VU n C T only for isometric processes

1V

V

dQC

n dT

Non-ideal gas:

Isobaric Processes & Specific Heat

Isobaric Process : constant P

2 1W p V V p V

Q U W U p V

isobaric processesPQ n C T

CP = molar specific heat at constant pressure

P Vn C T n C T p V Ideal gas, isobaric :

Vn C T n R T

P VC C R Ideal gas

Isotherms

1P

P

dQC

n dT

Adiabatic Processes

Adiabatic process: Q = constant

e.g., insulated system, quick changes like combustion, …

U W

Tactics 18.1. p V const adiabat, ideal gas

1P

V

C

C

1T V const Prob. 66

1 1 2 2

1

p V p VW

Prob. 62

Adiabatic: larger p

cdf

TACTIC 18.1. Adiabatic Equation

Ideal gas, any process: VdU n C dT

p dV

p V n R T

Adiabatic process: dU dWVn C dT

p dV V d p n R dT

V

p p dV V dpdV

C R

0V VR C p dV C V d p

0p VC p dV C V d p

0dV d p

V p

p

V

C

C

ln lnV p const ln p V p V const

Example 18.3. Diesel Power

Fuel ignites in a diesel engine from the heat of compression (no spark plug needed).

Compression is fast enough to be adiabatic.

If the ignit temperature is 500C, what compression ratio Vmax / Vmin is needed?

Air’s specific heat ratio is = 1.4, & before the compression the air is at 20 C.

1T V const

1 / 1.4 1273 500

273 20

K K

K K

1 / 1

max min

min max

V T

V T

11

Ideal Gas Processes

Cyclic Processes

Cyclic Process : system returns to same thermodynamic state periodically.

Example 18.4. Finding the Work

An ideal gas with = 1.4 occupies 4.0 L at 300 K & 100 kPa

pressure.

It’s compressed adiabatically to ¼ of original volume,

then cooled at constant V back to 300 K,

& finally allowed to expand isothermally to its original V.

How much work is done on the gas?

1A A B B

AB

p V p VW

741 J

AB (adiabatic):

0BCW BC (isometric):

ln ACA

C

VW n R T

VCA (isothermal):

1.4 1100 4.0 1 4

1.4 1

kPa L

AB A

B

Vp p

V

1

11

A A AAB

B

p V VW

V

ln 4A Ap V 555 J

work done by gas: ABCA AB BC CAW W W W 186 J

18.3. Specific Heats of an Ideal Gas

3

2ideal gasU N k T 1

V

UC

n T

Ideal gas: 21

2K m v

3

2k T

3

2n R T

3

2R

5

2RP VC C R P

V

C

C

5

3 1.67

Experimental values ( room T ):

For monatomic gases, 5/3, e.g., He, Ne, Ar, ….

For diatomic gases, 7/5 = 1.4, CV = 5R/2, e.g., H2 , O2 , N2 ,

….

For tri-atomic gases, 1.3, CV = 3.4R, e.g., SO2 , NO2 , ….

Degrees of freedom (DoF) = number of independent

coordinates required to describe the system

Single atom: DoF = 3 (transl)

For low T ( vib modes not active ) :

Rigid diatomic molecule : DoF = 5 (3 transl + 2 rot)

Rigid triatomic molecule : DoF = 6 (3 transl + 3 rot)

The Equipartition Theorem

Equipartition theorem ( kinetic energy version):

For a system in thermodynamic equilibrium, each degree of freedom of a

rigid molecule contributes ½ kT to its average energy.

Equipartition theorem ( general version):

For a system in thermodynamic equilibrium, each degree of freedom described

by a quadratic term in the energy contributes ½ kT to its average energy.

2A

fU n N k T

2V

fC R

2

fn R T

2f

f

12P

fC R

DoF ( f ) CV CP

Monatomic 3 3/2 5/2 5/3

Diatomic 5 5/2 7/2 7/5

Triatomic 6 3 4 4/3

Example 18.5. Gas Mixture

A gas mixture consists of 2.0 mol of oxygen (O2) & 1.0 mol of Argon (Ar).

Find the volume specific heat of the mixture.

2.2 R

2 2

5

2O OU n R T3

2Ar ArU n R T

2

5 3

2 2mix O ArU n n R T

1 mixmix

UC

n T

5 32.0 1.0

2 22.0 1.0

mol molR

mol mol

2

2

5 32 2O Ar

O Ar

n nR

n n

Quantum Effects

CV of H2 gas as function of T.

Below 20 K hydrogen is liquid,

above 3200 K it dissociates into individual atoms.

Quantum effect:

Each mechanism has a threshold energy.

Etransl < Erot < Evib

Translation

rotation+Translation

rotation+Translation+vibration

RepriseQuasi-static process : Arbitrarily slow process such that system always stays arbitrarily close to thermodynamic equilibrium.

Reversible process: Any changes induced by the process in the universe (system + environment) can be removed by retracing its path.

a c : Free expansion with no dissipative work.c b : Adiabatic.

a d : Adiabatic.d b : Free expansion with no dissipative work.

a e : Adiabatic.e b : Adiabatic dissipative work.

Insulated gas

1st law: The net adiabatic work done in all 3 processes are equal (shaded areas are equal).

Dissipative work: Work done on system without changing its configuration, irreversible.