Mayor’s Relation : Cp – Cv = R
Note: C of a gas depends on the process of that gas, (which is infinite in Isothermal Process).
Ratio of specific heat of gasses :
$ \displaystyle \gamma = \frac{C_P}{C_V} = (1 + \frac{2}{f})$
Cp – Cv = R
$ \displaystyle C_V = \frac{R}{\gamma – 1} $
$ \displaystyle C_P = \frac{\gamma R}{\gamma – 1} $
Indicator Diagram:
This is graph between pressure and volume of a system under going operation,
(1) Every point of Indicator Diagram represents a unique state (P, V, T) of gases.
(2) Every curve on Indicator Diagram represents a unique process.
Thermodynamic Processes :
Isochoric Process (V = constant)
dV = 0
⇒ dW = 0
By First Law of Thermodynamic
dQ = dU = n Cv dT
$ \displaystyle Q = \int_{T_1}^{T_2}n C_V dT = n C_V(T_2 – T_1) $
Isobaric Process (P = constant)
dP = 0
By First Law of Thermodynamics
dQ = dU + dW
$ \displaystyle n C_P(T_2 – T_1) = \frac{f}{2}n R(T_2 – T_1)+ n R(T_2 – T_1) $
where dW = n R (T2-T1 )
* Be careful if ΔV = 0 then not necessarily an Isochoric Process.
* If ΔP = 0 then not necessarily an Isobaric Process
Isothermal Process (T = constant):
dU = 0 (∴ dT = 0 )
PV = K
By First Law of Thermodynamics
$ \displaystyle \int dQ = \int dW $
$ \displaystyle \int dQ = \int P dV $
$ \displaystyle Q= W = nRT\int_{V_1}^{V_2}\frac{dV}{V} $
$ \displaystyle W = nRT ln\frac{V_2}{V_1} $
$ \displaystyle W = nRT ln\frac{P_1}{P_2} $
Adiabatic Process
dQ = 0
but , if ΔQ = 0 , it is not necessarily adiabatic.
∴ dW = – dU , By First Law of Thermodynamics
$ \displaystyle W = -\int_{T_1}^{T_2}\frac{nR dT}{\gamma – 1}$
$ \displaystyle W =\frac{n R(T_1 – T_2)}{\gamma – 1} $
$ \displaystyle W =\frac{(P_1 V_1 – P_2 V_2)}{\gamma – 1} $
How to get the process Equation for adiabatic
(i) First Law of Thermodynamics with process condition
$ \displaystyle dU = -dW = \frac{nRdT}{\gamma – 1} $
(ii) Differential form of gas law
d(PV) = d(nRT)
PdV + V dP = n R dT
$ \displaystyle dW = PdV = -\frac{nRdT}{\gamma – 1} $
P dV + V dP = -(γ-1)PdV
V dP = -(γPdV)
$ \displaystyle \frac{dP}{P} = -\gamma\frac{dV}{V} $
lnP = -γlnV + lnC
PVγ = constant
TVγ-1 = constant
TγP1-γ= constant
For Adiabatic Process PVγ = constant
$ \displaystyle |\frac{dP}{dV} |_{adiabatic} = \gamma |\frac{dP}{dV}|_{isothermal} $
Slope of adiabatic curve is more in magnitude in comparison to the slope of the isothermal curve.
Bulk Modulus of Gases:
$ \displaystyle B = -\frac{\Delta P}{\Delta V/V} $
Isothermal bulk modulus of Elasticity
$ \displaystyle E_T = -\frac{dP}{dV/V} = – V(\frac{\partial P}{\partial V})_{isothermal} $
Adiabatic bulk modulus of Elasticity
$ \displaystyle E_{adia} = -\frac{dP}{dV/V} = – V(\frac{\partial P}{\partial V})_{adiabatic} $