Mayor’s Relation , Thermodynamic Processes

Mayor’s Relation : C_p – C_v = 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})$

C_p – C_v = 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 C_v 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

PV^γ = constant

TV^γ-1 = constant

T^γP^1-γ= constant

For Adiabatic Process PV^γ = 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} $

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