please explain cp-cv=R briefly

Relationship between Cv and Cp for an ideal gas is as follows:

Heat capacity, C=q/T   [where, q: Heat; T: Temperature]

At constant volume, Heat capacity is denoted by Cv.

So, Cv= qv/∆T

Or, qv=Cv∆T  -----------------(1)

Since, ∆U=q - P∆V  [Where, ∆U: Internal energy change]

At constant volume , P∆V=0 

Therefore, at constant volume, ∆U=qv  -------(2)

From (1) and (2),

∆U =Cv∆T 

At constant pressure, Heat capacity is denoted by Cp.

So, Cp= qp/∆T

qp= Cp∆T  ---------------------(3)

we know, ∆H=∆U+∆(PV)   [where, ∆H:enthalpy change]

At constant pressure, ∆H=∆U+ P∆V

Since at constant pressure, ∆U=qp- P∆V

qp=∆U+P∆V 

qp=∆H  -------------(4)

From (3) and (4),

∆H= Cp∆T

Since, ∆H=∆U+∆(PV)

  ∆H=∆U+∆(nRT)  [we are deriving Relationship between Cv and Cp for an ideal gas]

  ∆H=∆U+∆(RT)  [For 1 mole of an ideal gas]

  ∆H=∆U+R∆T

After putting the values of ∆H and ∆U in above equation ,we get,

Cp∆T = Cv∆T + R∆T

 Cp = Cv + R or Cp- Cv = R

 

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