Comparison of the Slopes and Work done of an Isothermal and an Adiabatic Curve

Comparison of the Slopes and Work done of an Isothermal and an Adiabatic Curve

Comparison of the Slopes of an Isothermal and an Adiabatic Curve

Isothermal Process : 

For an isothermal process,

$PV = \text{constant}$

Differentiating,

$PdV + VdP = 0$

$\left(\frac{dP}{dV}\right)_{\text{iso}} = -\frac{P}{V}$

This represents the slope of the isothermal curve.

Adiabatic Process

For an adiabatic process,

$PV^\gamma = \text{constant}$

Differentiating,

$P\,\gamma V^{\gamma - 1} dV + V^\gamma dP = 0$

$\left(\frac{dP}{dV}\right)_{\text{adi}} = -\gamma \frac{P}{V}$

Comparison : 

From equations above,

$\left(\frac{dP}{dV}\right)_{\text{adi}} = \gamma \left(\frac{dP}{dV}\right)_{\text{iso}}$

Since $(\gamma > 1 )$, therefore,

$\left(\frac{dP}{dV}\right)_{\text{adi}} > \left(\frac{dP}{dV}\right)_{\text{iso}}$

Conclusion: 

The slope of the adiabatic curve is steeper than that of the isothermal curve.

Comparison of Work Done During Isothermal and Adiabatic Processes 

Show that work done during isothermal expansion is greater than during adiabatic expansion of a gas, and work done during adiabatic compression is greater than during isothermal compression.

(i) Expansion

Isothermal and adiabatic expansions between the same initial volume $( V_1 )$ and final volume $( V_2)$ are represented by the curves MI and MA respectively.

Work done during isothermal expansion:

$W_{iso}$= $\int_{V_1}^{V_2} PdV$ 

$W_{iso}$= Area under isothermal curve (MI)

$W_{iso}$= Area $MIV_2V_1$

Work done during adiabatic expansion:

$W_{adi}$= $\int_{V_1}^{V_2} PdV$

$W_{adi}$= Area under adiabatic curve (MA)

$W_{adi}$= Area $MAV_2V_1$

Since Area $MIV_2V_1$ > Area $MAV_2V_1$

we have:

$W_{iso}$ > $W_{adi}$

Conclusion:Work done during isothermal expansion is greater than that during adiabatic expansion for the same change in volume.

(ii) Compression : 

Isothermal and adiabatic compressions between the same initial volume $( V_1)$ and final volume $( V_2)$ are represented by the curves MI and MA respectively.

Work done during isothermal compression given by 

$W_{iso}$ = Area under isothermal curve (MI)= Area $MIV_1V_2$

Work done during adiabatic compression:

$W_{adi}$ = Area under adiabatic curve (MA)= Area $MAV_1V_2$

Since , Area $MAV_1V_2$ > Area $MIV_1V_2$

we have:

$W_{adi}$ > $W_{iso}$

Conclusion: 

Work done during adiabatic compression is greater than that during isothermal compression for the same change in volume.