Basic Mechanical Engineering (3110006)

Let,
${\mathrm{T}}_1 , {\mathrm{T}}_2$ = Initial and final temperature of gas respectively,  $\mathrm{K}$
${\mathrm{V}}_1 , {\mathrm{V}}_2$ = Initial and final volume of gas respectively, ${\mathrm{m}}^3$ 
${\mathrm{C}}_{\mathit{p}} , {\mathrm{C}}_{\mathrm{v}}$ = Specific heat at constant pessure & constant volume respectively, $\mathrm{kJ} / \mathrm{Kg} \mathrm{K}$ 
$\mathrm{R}$ = Characteristic gas constant in $\mathrm{kJ} / \mathrm{Kg} \mathrm{K}$ 
$\mathrm{p}$ = Pressure of gas, $\mathrm{kPa}$ 

Case-(1) Constant volume process

Consider m kg of gas heated at constant volume from  ${\mathrm{T}}_1$ to ${\mathrm{T}}_2$. 
Heat supplied at constant volume is,
$\mathrm{Q} = {\mathrm{mC}}_{\mathrm{v}} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right) .........[ 1 ]$ 

Apply 1st law of thermodynamics,
$\mathrm{Q} = ∆\mathrm{U} + \mathrm{W}$ 

But, W is zero for constant volume process.
So, $\mathrm{Q} = ∆\mathrm{U}$ 

From Eq. [1],
$∆\mathrm{U} = {\mathrm{mC}}_{\mathrm{v}} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right) .........[ 2 ]$ 

Case-(2) Constant pressure process

Heat supplied at constant pressure is,
$\mathrm{Q} = {\mathrm{mC}}_{\mathrm{p}} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right) .........[ 3 ]$ 

Work done $\mathrm{W} = \mathrm{p} \left( {\mathrm{V}}_2 - {\mathrm{V}}_1 \right) .........[ 4 ]$ 

Apply 1st law of thermodynamics,
$\mathrm{Q} = ∆\mathrm{U} + \mathrm{W}$ 

from Eq. [ 2 ], [ 3 ] & [ 4 ],

$\Rightarrow {\mathrm{mC}}_{\mathrm{p}} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right) = {\mathrm{mC}}_{\mathrm{v}} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right) + \mathrm{p} \left( {\mathrm{V}}_2 - {\mathrm{V}}_1 \right)$ 

Apply characteristic gas equation, ${\mathrm{pV}}_1 = {\mathrm{mRT}}_1$ and ${\mathrm{pV}}_2 = {\mathrm{mRT}}_2$ in above equation, 

$\Rightarrow {\mathrm{mC}}_{\mathrm{p}} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right) = {\mathrm{mC}}_{\mathrm{v}} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right) + \mathrm{mR} \left( {\mathrm{T}}_2 - {\mathrm{T}}_1 \right)$ 

$\Rightarrow {\mathrm{C}}_{\mathrm{p}} = {\mathrm{C}}_{\mathrm{v}} + \mathrm{R}$ $\Rightarrow {\mathrm{C}}_{\mathrm{p}} - {\mathrm{C}}_{\mathrm{v}} = \mathrm{R}$