Properties of Ideal gases ∣ Charactaristic of gas equation ∣ Specific Heat
Ideal gases
The term gas is applied to a particular phase of a pure substance which will fills the system boundary, and no change of phase takes place or is contemplated. They always exist in gaseous form. For this reason, they have been called permanent gases.
Perfect or ideal gases are characterised by the absence of forces between the molecules, and volume occupied by the molecules is infinitely small compared with that of gas. They remain gaseous form and flows the gas laws over a wide range of conditions.
Boyle and Charles investigated the behaviour of gases and formulated the laws.
Boyle’s law states that if a perfect gas undergoes a change of state at constant temperature,
then
p ∝ (1/ʋ)T = constant ……… (i)
pʋ = constant or p₁ʋ₁ = p₂ʋ₂
Charles law states that for a perfect gas undergoing a change of state at constant pressure,
ʋ ∝ (T)p = constant ………(ii)
(ʋ/T) = constant
or (ʋ₁/T₁) = (ʋ₂/T₂)
Also, at constant volume,
p ∝ (T)ʋ=constant (Gay-lussac law) …..(iii)
(p/T) = constant
or (p₁/T₁) = (p₂/T₂)
where p = Pressure, ʋ = specific volume and T = temperature
Characteristic gas equation
A relation between p, ʋ and T may be obtained by combining Boyle’s law and Charles law. Combining equation (i) and (ii)
ʋ ∝ (T/p)
(pʋ/T) = constant
Thus, (p₁ʋ₁/T₁) = (p₂ʋ₂/T₂) = constant
If the constant is represented by R which is called characteristic gas constant, then
(pʋ/T) = R
or pʋ = RT −Characteristic gas equation
where ʋ = specific volume m³/kg
= V/m
∴ For given mass, m of gas, the gas equation becomes
pV = mRT
where R = characteristic gas constant
= (pV/mT) = (N/m²)m³/kg K = (Nm/kg K) or (J/kg K)
Specific heat
The specific heat of a substance is defined as the amount of heat required to raise the temperature of unit mass through one degree.
Specific heat, C = (dQ/dT)
Heat capacity :
The amount of heat required to raise the temperature of an object of one degree is called heat capacity
Specific heat for gases :
Specific heat at constant volume,
Cʋ = (dQ/dT)ʋ = constant = (du/dT) or du = Cʋ.dT
Between two states,
u₂ − u₁ = Cʋ (T₂ − T₁)
For a given mass, m kg of gas
U₂ − U₁ = mCʋ (T₂ − T₁)
Specific heat at constant pressure
Cp = (dQ/dT)p=constant = (dh/dT) or dh = Cp.dT
Between initial and final states,
h₂ − h₁ = Cp(T₂ − T₁)
For a given mass, m kg of a gas,
H₂ − H₁ = m.Cp(T₂ − T₁)
The term (H₂ − H₁) is called change in enthalpy which is equal to heat transferred at constant pressure
Relation between specific heats and gas constant
The specific heats at constant volume Cʋ and at constant pressure, Cp are defined by the equations
Cʋ = (du/dT) and Cp = (dh/dT)
Also, enthalpy is a property which is defined by the equation,
h = u + pV
h = u + RT or dh = du + R dT
or (dh/dT) = (du/dT) + R
∴ Cp = Cʋ + R …….(i)
The ratio, (Cp/Cʋ) is called adiabatic index (γ). In terms of adiabatic index the equation (i) can be expressed.
γ = 1 + R/Cʋ
or Cʋ = R/(γ – 1) and Cp = (Rγ)/(γ – 1)
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