Thermo Formulae

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COMMON FORMULAS FOR THERMODYNAMICS

All relations are for unit mass (or unit mole) of material

S. No. Formula CommentsLAWS

First Law: dU = dQ + dW ; ∆U = Q + W Differential and Integral Forms

Second Law: ∆S Total ≥ 0 Total means System + Surroundings

dU = TdS + PdV 1st & 2nd Laws Mixed ( Reversible Process)

dH = TdS + VdP 1st & 2nd Laws Mixed ( Reversible Process)

( ) ; ( )dH dU d PV H U PV ≡ + Δ ≡ Δ + Δ Definition of Enthalpy

; p p

dH C dT H C dT = Δ = ∫ Enthalpy calculation for ideal gas. (May be used

for real gases, liquid and solids with caution).

;v v

dU C dT U C dT = Δ = ∫ Internal Energy calculation for ideal gas. (May be

used for real gases, liquid and solids with caution).

;dW P dV W P dV = − = − ∫ Work calculation for reversible process.Efficiency of Carnot Engine:

1 Net C

H

W T

Q T = −

T H and T C are temperatures of hot and cold

reservoirs respectively

; R RdQ dQ

dS S T T

≡ Δ ≡ ∫ Definition of Entropy Change. Valid for all

reversible process with or without work.

;dQ dQ

dS S T T

= Δ = ∫ Entropy Change for all processes (reversible or irreversible) involving heat transfer (Q) only i.e. no

work involved.

QS

T Δ =

Entropy Change for all processes (reversible or

irreversible) involving heat transfer (Q) only if the

heating / cooling is at constant temperature. ∆S Total = 0 If all processes (isothermal, adiabatic, etc.)

between the system and surroundings are

reversible.

∆S Total > 0 If any one or more processes (isothermal,

adiabatic, etc.) between the system and surroundings are irreversible

∆S = 0 For any reversible as well as adiabatic process

Special Ideal Gas Processes

Isometric Process (∆V = 0)

• W = 0

• v

Q C dT = ∫

• Either reversible or irreversible process


Isobaric Process ((∆ P = 0)

• W P V = − Δ

• P

Q C dT = ∫

• Reversible process only


Isothermal Process (∆T = 0)

• 0 ; 0U H Δ = Δ =

• Q W PdV = − = − ∫

• 2 2

1 1

ln lnV P

W RT RT V P

= − =




Sign Conventions: Q & W : +ve if ENTER the system

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Adiabatic Process (Q = 0)

• V dW dU C dT = =

• 1v

v

RW C T

C = Δ = +

• PV = Constant ; P

V

C

C γ ≡

•

11

2 1 2

1 2 1

T V P

T V P

γ γ

γ

−−

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

•

1 1

1 1 12 21 1

1 1 1 1

PV RT P P W

P P

γ γ

γ γ

γ γ

− −⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥= − = −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥− −⎝ ⎠ ⎝ ⎠

⎣ ⎦ ⎣ ⎦

• Either reversible or irreversible.

• Reversible process with constant Cv




Isentropic Process (∆S = 0) All correlations for adiabatic process are

applicable with additional constraint of reversibility.

Equations of State etc.:

Ideal Gas: PV = RT May be used for real gases with caution.

Units of R to be used with care.

Real Gases: PV = ZRT Many other relations also exist. Z is given as a

function of reduced temperature and pressure.Steam Tables Preferred choice for properties of water and steam.

Correlations and softwares are also available in

addition to tabulated values.

C P = C V + R For ideal gas (or nearly so)

C P , C V , and R must be used in molar units.

Liquids and Solids:

CP≈ CV

∆ H ≈ ∆U

Due to very low compressibilities

Values of Gas Constant R

Units Value

atm-ft³/(lb mol-°R) 0.73024

atm-l/(g mol-K) 0.08206

bar-l/(g mol-K) 0.08314472

Btu/(lb mol-°R) 1.9859

cal/(mol-K) 1.9859

hp-h/(lb mol-°R) 0.0007805

J/(g mol-K) 8.314472

kWh/(lb mol-°R) 0.000582

lbf-ft/(lb mol-°R) 1545.349

mmHg-l/(g mol-K) 62.364

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Thermo Formulae