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$dQ = dU + dW$;$dQ = dU + dW = dU + PdV$,
At therefore, at constant Volume, Work done by the gas is zero, so $dQ = dU$;
 
$dQ = f/2 nRdT;$&
  
$dQ/dT = f/2 nR $;From Law Of Equipartition Of Energy,
  
$nCvdT = f/2 nRdT $;for 'n' moles of an ideal gas
$Cv = f/2R$;$dU = \frac{f}{2}nRdT$

$\implies$
  
$nCpdT = f/2 nRdT + PdV$$dU = dQ = \frac{f}{2}nRdT$ (at constant volume)
$nC_VdT = \frac{f}{2}nRdT$
hence,
$C_V = \frac{f}{2}R$;

also,
$nCpdT = \frac{f}{2}nRdT + PdV$ (Atat constant Pressure)
 

[$PdV +VdP = nRdT$;  (Differentiating $PV = nRT$)
 ]

$nCpdT = \frac{f}{2}nRdT + nRdT$
$nCpdT = f/2nRdT + nRdT$
$Cp = \frac{f}{2}R + R$
$Cp = f/2 R + R$
hence,
$Cp = Cv + R$
  

$dQ = dU + dW$;
At constant Volume, Work done by the gas is zero so $dQ = dU$;
$dQ = f/2 nRdT;$
  $dQ/dT = f/2 nR $;
  $nCvdT = f/2 nRdT $;
$Cv = f/2R$;
  $nCpdT = f/2 nRdT + PdV$;(At constant Pressure)
 $PdV +VdP = nRdT$;(Differentiating $PV = nRT$)
  $nCpdT = f/2nRdT + nRdT$
$Cp = f/2 R + R$
$Cp = Cv + R$
 

$dQ = dU + dW = dU + PdV$,
therefore, at constant Volume, Work done by the gas is zero, so $dQ = dU$ 
&  
From Law Of Equipartition Of Energy,  
for 'n' moles of an ideal gas $dU = \frac{f}{2}nRdT$

$\implies$  
$dU = dQ = \frac{f}{2}nRdT$ (at constant volume)
$nC_VdT = \frac{f}{2}nRdT$
hence,
$C_V = \frac{f}{2}R$;

also,
$nCpdT = \frac{f}{2}nRdT + PdV$ (at constant Pressure)

[$PdV +VdP = nRdT$;  (Differentiating $PV = nRT$)]

$nCpdT = \frac{f}{2}nRdT + nRdT$
$Cp = \frac{f}{2}R + R$
hence,
$Cp = Cv + R$ 

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$dQ = dU + dW$; At
At constant Volume, Work done by the gas is zero so dQ = dU;$dQ = dU$;
$nCvdT = f/2 nRdT $;$dQ = f/2 nRdT;$
$Cv = f/2R$
$dQ/dT = f/2 nR $;
$dQ/dT = f/2 R $$nCvdT = f/2 nRdT $;
$Cv = f/2R$;
$nCpdT = f/2 nRdT + PdV$;(At constant Pressure)
$PdV +VdP = nRdT$;(Differentiating $PV = nRT$)
$nCpdT = f/2nRdT +PdV$(At constant Pressure)
$nCpdT = f/2nRdT + nRdT$
$Cp = f/2 R + R$
$Cp = Cv + R$

$dQ = dU + dW$; At constant Volume, Work done by the gas is zero so dQ = dU;
$nCvdT = f/2 nRdT $;
$Cv = f/2R$;
$dQ/dT = f/2 R $;
$nCpdT = f/2 nRdT + PdV$;(At constant Pressure)
$PdV +VdP = nRdT$;(Differentiating $PV = nRT$)
$nCpdT = f/2nRdT +PdV$(At constant Pressure)
$nCpdT = f/2nRdT + nRdT$
$Cp = f/2 R + R$
$Cp = Cv + R$

$dQ = dU + dW$;
At constant Volume, Work done by the gas is zero so $dQ = dU$;
$dQ = f/2 nRdT;$
$dQ/dT = f/2 nR $;
$nCvdT = f/2 nRdT $;
$Cv = f/2R$;
$nCpdT = f/2 nRdT + PdV$;(At constant Pressure)
$PdV +VdP = nRdT$;(Differentiating $PV = nRT$)
$nCpdT = f/2nRdT + nRdT$
$Cp = f/2 R + R$
$Cp = Cv + R$

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dQ = dU + dW;$dQ = dU + dW$; At constant Volume, Work done by the gas is zero so dQ = dU;
nCvdT = f/2 nRdT ; Cv = f/2R;$nCvdT = f/2 nRdT $;
$Cv = f/2R$;
dQ/dT = f/2 nR ;$dQ/dT = f/2 R $;
$nCpdT = f/2 nRdT + PdV$;(At constant pressure; nCpdT = f/2 nRdT + PdV;Pressure)
PdV +VdP = nRdT;$PdV +VdP = nRdT$;(Differentiating PV = nRT$PV = nRT$)
nCpdT = f/2nRdT +PdV$nCpdT = f/2nRdT +PdV$(At constant Pressure)
nCpdT = f/2nRdT + nRdT$nCpdT = f/2nRdT + nRdT$
Cp = f/2 R + R$Cp = f/2 R + R$
Cp = Cv + R$Cp = Cv + R$

dQ = dU + dW; At constant Volume, Work done by the gas is zero so dQ = dU;
nCvdT = f/2 nRdT ; Cv = f/2R;
dQ/dT = f/2 nR ;
At constant pressure; nCpdT = f/2 nRdT + PdV;
PdV +VdP = nRdT;(Differentiating PV = nRT)
nCpdT = f/2nRdT +PdV(At constant Pressure)
nCpdT = f/2nRdT + nRdT
Cp = f/2 R + R
Cp = Cv + R

$dQ = dU + dW$; At constant Volume, Work done by the gas is zero so dQ = dU;
$nCvdT = f/2 nRdT $;
$Cv = f/2R$;
$dQ/dT = f/2 R $;
$nCpdT = f/2 nRdT + PdV$;(At constant Pressure)
$PdV +VdP = nRdT$;(Differentiating $PV = nRT$)
$nCpdT = f/2nRdT +PdV$(At constant Pressure)
$nCpdT = f/2nRdT + nRdT$
$Cp = f/2 R + R$
$Cp = Cv + R$

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