# 12.2 The first law of thermodynamics and some simple processes  (Page 3/12)

 Page 3 / 12

## Total work done in a cyclical process equals the area inside the closed loop on a PV Diagram

Calculate the total work done in the cyclical process ABCDA shown in [link] (b) by the following two methods to verify that work equals the area inside the closed loop on the $\text{PV}$ diagram. (Take the data in the figure to be precise to three significant figures.) (a) Calculate the work done along each segment of the path and add these values to get the total work. (b) Calculate the area inside the rectangle ABCDA.

Strategy

To find the work along any path on a $\text{PV}$ diagram, you use the fact that work is pressure times change in volume, or $W=P\Delta V$ . So in part (a), this value is calculated for each leg of the path around the closed loop.

Solution for (a)

The work along path AB is

$\begin{array}{lll}{W}_{\text{AB}}& =& {P}_{\text{AB}}\Delta {V}_{\text{AB}}\\ & =& \left(1\text{.}\text{50}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\right)\left(5\text{.}\text{00}×{\text{10}}^{–4}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\right)=\text{750}\phantom{\rule{0.25em}{0ex}}\text{J.}\end{array}$

Since the path BC is isochoric, $\Delta {V}_{\text{BC}}=0$ , and so ${W}_{\text{BC}}=0$ . The work along path CD is negative, since $\Delta {V}_{\text{CD}}$ is negative (the volume decreases). The work is

$\begin{array}{lll}{W}_{\text{CD}}& =& {P}_{\text{CD}}\Delta {V}_{\text{CD}}\\ & =& \left(2\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\right)\left(–5\text{.}\text{00}×{\text{10}}^{–4}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\right)\phantom{\rule{0.25em}{0ex}}\text{=}\phantom{\rule{0.25em}{0ex}}\text{–}\text{100}\phantom{\rule{0.25em}{0ex}}\text{J}\text{.}\end{array}$

Again, since the path DA is isochoric, $\Delta {V}_{\text{DA}}=0$ , and so ${W}_{\text{DA}}=0$ . Now the total work is

$\begin{array}{lll}W& =& {W}_{\text{AB}}+{W}_{\text{BC}}+{W}_{\text{CD}}+{W}_{\text{DA}}\\ & =& \text{750 J}+0+\left(-\text{100}\text{J}\right)+0=\text{650 J.}\end{array}$

Solution for (b)

The area inside the rectangle is its height times its width, or

$\begin{array}{lll}\text{area}& =& \left({P}_{\text{AB}}-{P}_{\text{CD}}\right)\Delta V\\ & =& \left[\left(\text{1.50}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\right)-\left(2\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}\right)\right]\left(5\text{.}\text{00}×{\text{10}}^{-4}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\right)\\ & =& \text{650 J.}\end{array}$

Thus,

$\text{area}=\text{650}\phantom{\rule{0.25em}{0ex}}\text{J}=W\text{.}$

Discussion

The result, as anticipated, is that the area inside the closed loop equals the work done. The area is often easier to calculate than is the work done along each path. It is also convenient to visualize the area inside different curves on $\text{PV}$ diagrams in order to see which processes might produce the most work. Recall that work can be done to the system, or by the system, depending on the sign of $W$ . A positive $W$ is work that is done by the system on the outside environment; a negative $W$ represents work done by the environment on the system.

[link] (a) shows two other important processes on a $\text{PV}$ diagram. For comparison, both are shown starting from the same point A. The upper curve ending at point B is an isothermal process—that is, one in which temperature is kept constant. If the gas behaves like an ideal gas, as is often the case, and if no phase change occurs, then $\text{PV}=\text{nRT}$ . Since $T$ is constant, $\text{PV}$ is a constant for an isothermal process. We ordinarily expect the temperature of a gas to decrease as it expands, and so we correctly suspect that heat transfer must occur from the surroundings to the gas to keep the temperature constant during an isothermal expansion. To show this more rigorously for the special case of a monatomic ideal gas, we note that the average kinetic energy of an atom in such a gas is given by

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
How we can toraidal magnetic field
How we can create polaidal magnetic field
4
Because I'm writing a report and I would like to be really precise for the references
where did you find the research and the first image (ECG and Blood pressure synchronized)? Thank you!!