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A simple case of work on a system of gases can be explained well by mechanical work. This deals with the pressure volume changes. For example work of expansion in case of reversible processes where there may be infinitesimally small changes of pressure or volume can be given by the following general expression
W = -
The following are the expressions for the work of expansion of an isothermal irreversible , reversible and adiabatic changes.
  1. Isothermal irreversible change= q= -w = Pex (Vf - Vi)
  2. Isothermal reversible change = q= -w = nRT ln
  3. Adiabatic change, q=o, Δ U = wad
Relation between enthalpy of a system at constant pressure (Δ H) and constant volume (Δ U) among the three physical states of matter:
H = U+ PV

Let the initial and final values of internal energies of a system be U1 ,U2 and the corresponding values of enthalpies be H1, H2 respectively. Let the volume changes be V1, V2 at constant pressure p. The above equation can be reformulated as

H2 -H1 = U2 -U1 + P(V2 -V1)
ΔH = ΔU + P ΔV

Since the P ΔV changes are nor appreciable in solids and liquids it is negligible and ΔH = ΔU
But in the case of gases P ΔV changes are appreciable and should be calculated.
Assuming the gases to behave ideally at constant pressure , we have
PV = nRT

Assuming initial(reactants) and final states(products)
P(Vp - Vr) = (np - nr) RT
P Δ V = Δng RT

hence for gases ,Δ H =Δ U +Δ ng RT
qp = Δ U + Δng RT ( Δ H = qp)

Intensive and Extensive properties:
The properties of a system which are independent of amount of the constituent particles are called intensive properties (temperature, refractive index, viscosity, density etc.). Those properties which depend upon the mass are extensive properties (enthalpy, volume, internal energy, heat capacity, no of moles etc.).

Heat Capacity and Meyer's relation:
Heat capacity of a system is defined as the quantity of heat required to raise the temperature of unit mass of substance through 1oc which can be expressed as q = c m ΔT = C ΔT

Quantity of heat at constant pressure = qp = CpΔT = ΔH
Quantity of heat at constant volume = qv = Cv ΔT = ΔU
For one mole of an ideal gas , ΔH = ΔU + Δ(pV)
CpΔT = Cv ΔT + R ΔT
Cp = Cv + R (or) Cp - Cv = R (This expression is known as Meyer's relation.)

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