Question

Determine the total work done by the gas system following an expansion process as shown in figure

Answer 1

Answer:

Work is the energy required to move something against a force.

The energy of a system can change due to work and other forms of energy transfer such as heat.

Gases do expansion or compression work following the equation: work = − P Δ V \text {work} = -\text P\Delta \text V work=−PΔV.

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Answer 2

Determine the total work done by the gas system following an expansion process as shown in figure.

Total work done by gas, W  

$\text { Total work done by gas, } \mathrm{W}_{\text {Total }}=\mathrm{W}_{\mathrm{AB}}+\mathrm{W}_{\mathrm{BC}}+\mathrm{W}_{\mathrm{CA}}$

​  $\mathrm{W}_{\mathrm{AB}}=\mathrm{n} R \mathrm{~T} \ln \frac{4 \mathrm{~V}}{\mathrm{~V}}=2 \mathrm{n} \mathrm{RT} \ln 2=2 \mathrm{P} \mathrm{V} \ln 2$

$\text { Also } \mathrm{P}_{\mathrm{A}} \mathrm{V}_{\mathrm{A}}=\mathrm{P}_{\mathrm{B}} \mathrm{V}_{\mathrm{B}} \text { (As } \mathrm{AB} \text { is an isothermal process) }$

$\text { or, } \mathrm{P}_{\mathrm{B}}=\frac{\mathrm{P}_{\mathrm{A}} \mathrm{V}_{\mathrm{A}}}{\mathrm{V}_{\mathrm{B}}}=\frac{\mathrm{PV}}{4 \mathrm{~V}}=\frac{\mathrm{P}}{4}$

Also P  A  V  A =P  B V B   (As AB is an isothermal process)

=P  B  V  B   (As AB is an isothermal process)

or, P  B  =  V  B P  A ​ V  A

​  =  4V PV  =  4 P

​$\mathrm{W}_{\mathrm{BC}}=\frac{\mathrm{P}}{4}(\mathrm{~V}-4 \mathrm{~V})=-\frac{3 \mathrm{PV}}{4}$

$=2 \mathrm{P} \mathrm{V} \ln 2-\frac{3 \mathrm{PV}}{4}+0$

In the step BC, the pressure remains constant. Hence the work done is,

$\text { Hence, the work done by the gas is } 0.636 \mathrm{PV} \text {. }$