5.0 Graphs

Conversion of graphs from one axis to other axis.

Lets take an example

Convert the graph in $Fig.35$ to $P$-$V$, $V$-$T$ and $\rho $-$T$ graph for an ideal gas

Process $AB$ is an isothermal process with $T$ = constant and ${P_B} > {P_A}$.

$P$-$V$ graph : $P \propto \frac{1}{V}$ $ \Rightarrow P - V$ graph is a hyperbola with ${P_B} > {P_A}$ and ${V_B} < {V_A}$.

$V$-$T$ graph : $T$ = constant. Therefore, $V$-$T$ graph is a straight line parallel to $V$-axis with ${V_B} < {V_A}$.

$\rho $-$T$ graph : $\rho = \frac{{PM}}{{RT}}\ \ or\ \ \rho \propto P$

AS $T$ is constant. Therefore $\rho $-$T$ graph is a straight line parallel to $\rho $-axis with ${\rho _B} > {\rho _A}\ as\ {P_B} > {P_A}$.

Process $BC$ is an isobaric process with $P$ = constant and ${T_C} > {T_B}$.

$P$-$V$ graph : As $P$ is constant. Therefore, $P$-$V$ graph is a straight line parallel to $V$ axis with ${V_C} > {V_B}$ ( because $V \propto T$ in an adiabatic process).

$V$-$T$ graph : In Isobaric process $V \propto T \Rightarrow V - T$ graph is straight line passing through origin, with ${T_C} > {T_B}$ and ${V_C} > {V_B}$.

$\rho $-$T$ graph : $\rho \propto \frac{1}{T}$ (when $P$ = constant ), $\rho $-$T$ graph is a hyperbola with ${T_C} > {T_B}$ and ${\rho _C} < {\rho _B}$.

There is no need of discussing $C-D$ and $D-A$ processes. As they are opposite to $AB$ and $BC$ respectively.