Question

Show how the internal energy U varies with P in isothermal, isochoric, isobaric
and adiabatic process.

Answer 1

To Show:

Show how the internal energy U varies with P in isothermal, isochoric, isobaric.

Solution:

We know that-

Change in internal energy ,U for any process is ΔU = $nC_{v}$Δ$T$.

where,   $C_{v}=\dfrac{f}{2}R$

this implies;

$\gamma = \dfrac{2}{f}+1\\\dfrac{f}{2} = \dfrac{1}{\gamma - 1}$

therefore, for all process;

ΔU =     $\left(\dfrac{nR}{\gamma - 1}\right)$   ΔT

since, $PV = n R T$  this implies;

$\Delta U= \dfrac{ P V}{\gamma - 1}$

Now,

for isothermal (constant temperature) ;

$\Delta P = 0$

⇒ $\Delta U = 0$ , because system pressure does not change regularly during the process.

for isochoric (constant volume);

In Isochoric process , system volume is kept constant and pressure change takes place regularly.

$\Delta U = Q$

for isobaric (constant pressure);

When the pressure is kept constant then after applying heat system volume changes and does work on the environment.

$\Delta U = Q - W$

for Adiabatic (no heat added);

implies ; Q = 0

$\Delta U=-W$