# Thermodynamic Equations

### Thermodynamics Nomenclature:

 T = temperature (oK) V = volume of system (cubic metres) P or p = pressure at the boundary of the system and its environment, (in pascals) W = work done by or on a system (joules) Q = heat transfer in or out of a system (joules) q = specific heat transfer in or out of a system (joules per kg) U = internal energy of a system mainly contained in solid and liquid components (joules) u = specific internal energy of a system (joules per kg) H = enthalpy of a system (joules) h = specific enthalpy (joules per kg) S = entropy (joules per oK) s = specific entropy (joules per kg per oK) n = heat capacity ratio (= Cv/Cp)

### Thermodynamics Equations:

Thermodynamics equations can be difficult to understand. The following is a simpified summary where the term “system” can be equated to a steam locomotive’s cylinder:

The First Law of Thermodynamics (conservation of energy) can be expressed as “The increase in internal energy of a system = the heat supplied to the system minus the energy that flows out in the form of Work that the system performs on it environment” [ref Wikipedia]. In this case, the external “environment” is the locomotive’s piston, hence the definition can be formulated by the equation:

δU = Q – Wpiston ………… (1)

which may also be written:

dU = dQ – dWpiston ………. (1a)

However, the work done on a system (locomotive cylinder) by changing its volume is dW = p.dV, hence:

dU = dQ – p.dV …………. (1b)

If the process (steam expansion) is assumed to be adiabatic – i.e. with no heat transfer in or out, then dQ = 0, whence

dU = – p.dV ………………. (1c)

However Enthalpy (see separate page) is defined as the sum of a system’s internal energy plus the product of its pressure and volume – i.e.

H = U + P.V ……………..(2)

from which a change in enthalpy can be defined (by differentiation) as

dH = dU + p.dV + V.dp ……………..(2a)

Thus by combining equations (1c) and (2a) we get (for adiabatic expansion): dH = -p.dV + p.dV + V.dp, or

dH = V.dp ……………..(3)

Combining eqns (1c) and (3) gives:

dH/dU = – V.dP / P.dV …………….. (4)

The Second Law of Thermodynamics (heat always flows to regions of lower temperature) can be expressed as “a change in the entropy (S) of a system is the infinitesimal transfer of heat (Q) to a closed system driving a reversible process, divided by the equilibrium temperature (T) of the system” [ref Wikipedia].  This definition is formulated by the equation:

dS = δQ/T or dQ = T.dS ………………(5)

By combining Eqn (4) with (1b), a change in internal energy is given by:

dU = T.dS – p.dV ………………………….(6)

Ideal Gas Laws (from physics): The ideal gas law is defined by the equation:

pVn = k …………………………. (7)

where n is the “heat capacity ratio“: n = Cp / Cv = – V.dp / p.dV [ref Wikipedia]

Thus from equation (4):

n = dH/dU

————  page in progress as at 10th Mar 2011  ———–