# 3.2 Work, heat, and internal energy  (Page 2/6)

 Page 2 / 6
$W={\int }_{{V}_{1}}^{{V}_{2}}pdV={\int }_{{V}_{1}}^{{V}_{2}}\left(\frac{nRT}{V}\right)\phantom{\rule{0.2em}{0ex}}dV.$

The expansion is isothermal, so T remains constant over the entire process. Since n and R are also constant, the only variable in the integrand is V , so the work done by an ideal gas in an isothermal process is

$W=nRT{\int }_{{V}_{1}}^{{V}_{2}}\frac{dV}{V}=nRT\text{ln}\frac{{V}_{2}}{{V}_{1}}.$

Notice that if ${V}_{2}>{V}_{1}$ (expansion), W is positive, as expected.

The straight lines from A to B and then from B to C represent a different process. Here, a gas at a pressure ${p}_{1}$ first expands isobarically (constant pressure) and quasi-statically from ${V}_{1}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}{V}_{2}$ , after which it cools quasi-statically at the constant volume ${V}_{2}$ until its pressure drops to ${p}_{2}$ . From A to B , the pressure is constant at p , so the work over this part of the path is

$W={\int }_{{V}_{1}}^{{V}_{2}}pdV={p}_{1}{\int }_{{V}_{1}}^{{V}_{2}}dV={p}_{1}\left({V}_{2}-{V}_{1}\right).$

From B to C , there is no change in volume and therefore no work is done. The net work over the path ABC is then

$W={p}_{1}\left({V}_{2}-{V}_{1}\right)+0={p}_{1}\left({V}_{2}-{V}_{1}\right).$

A comparison of the expressions for the work done by the gas in the two processes of [link] shows that they are quite different. This illustrates a very important property of thermodynamic work: It is path dependent . We cannot determine the work done by a system as it goes from one equilibrium state to another unless we know its thermodynamic path. Different values of the work are associated with different paths.

## Isothermal expansion of a van der waals gas

Studies of a van der Waals gas require an adjustment to the ideal gas law that takes into consideration that gas molecules have a definite volume (see The Kinetic Theory of Gases ). One mole of a van der Waals gas has an equation of state

$\left(p+\frac{a}{{V}^{2}}\right)\left(V-b\right)=RT,$

where a and b are two parameters for a specific gas. Suppose the gas expands isothermally and quasi-statically from volume ${V}_{1}$ to volume ${V}_{2}.$ How much work is done by the gas during the expansion?

## Strategy

Because the equation of state is given, we can use [link] to express the pressure in terms of V and T . Furthermore, temperature T is a constant under the isothermal condition, so V becomes the only changing variable under the integral.

## Solution

To evaluate this integral, we must express p as a function of V . From the given equation of state, the gas pressure is

$p=\frac{RT}{V-b}-\frac{a}{{V}^{2}}.$

Because T is constant under the isothermal condition, the work done by 1 mol of a van der Waals gas in expanding from a volume ${V}_{1}$ to a volume ${V}_{2}$ is thus

$\begin{array}{cc}\hfill W& =\underset{{V}_{1}}{\overset{{V}_{2}}{\int }}\left(\frac{RT}{V-b}-\frac{a}{{V}^{2}}\right)={|RT\text{ln}\left(V-b\right)+\frac{a}{V}|}_{{V}_{1}}^{{V}_{2}}\hfill \\ & =RT\text{ln}\left(\frac{{V}_{2}-b}{{V}_{1}-b}\right)+a\left(\frac{1}{{V}_{2}}-\frac{1}{{V}_{1}}\right).\hfill \end{array}$

## Significance

By taking into account the volume of molecules, the expression for work is much more complex. If, however, we set $a=0$ and $b=0,$ we see that the expression for work matches exactly the work done by an isothermal process for one mole of an ideal gas.

Check Your Understanding How much work is done by the gas, as given in [link] , when it expands quasi-statically along the path ADC ?

${p}_{2}\left({V}_{2}-{V}_{1}\right)$

## Internal energy

The internal energy     ${E}_{\text{int}}$ of a thermodynamic system is, by definition, the sum of the mechanical energies of all the molecules or entities in the system. If the kinetic and potential energies of molecule i are ${K}_{i}$ and ${U}_{i},$ respectively, then the internal energy of the system is the average of the total mechanical energy of all the entities:

is this book for preparing IIT or neet?
is it possible to increase the temperature of a gas without adding heat to it?
I'm not sure about it, but I think it's possible. If you add some form of energy to the system, it's a possibility. Also, if you change the pression or the volume of the system, you'll increase the kinetic energy of the system, increasing the gas temperature. I don't know if I'm correct.
playdoh
For example, if you get a syringe and close the tip(sealing the air inside), and start pumping the plunger, you'll notice that it starts getting hot. Again, I'm not sure if I am correct.
playdoh
you are right for example an adiabatic process changes all variables without external energy to yield a temperature change. (Search Otto cycle)
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whate is fckg diagrame?
Arzoodan
why do we use integration?
To know surfaces below graphs.
Jan
To find a Primitive function. Primitive function: a function that is the origin of another
playdoh
yes
Dharmdev
what is laps rate
Dharmdev
Г=-dT/dZ that is simply defination
Arzoodan
what is z
Dharmdev
to find the area under a graph or to accumulate .e.g. sum of momentum over time is no etic energy.
Naod
Z is alt.,dZ altv difference
Arzoodan
what is the Elasticty
it is the property of the by virtue of it regains it's original shape after the removal of applied force (deforming force).
Prema
property of the material
Prema
which type of cable is suitable for patrol station wiring
what is calorimeter
heat measuring device
Suvransu
What is mean electric potential
Electric density formula
Int E•dA
Vineet
what is relation betweeen potential energy of a system and work done in bringing last chaege from infinity
Einstein and his general theory of relativity, a very nice concept,which revolutionize our modern world.
what's the theory of relativity?
Piyali
it related to universe time period
jyotirmayee
it is a theory in which time is taken as relative,for diff. motion of objects.
Antares
no !!!
jyotirmayee
it is E=mc'square
jyotirmayee
what is black hole?
jyotirmayee
E=mc^2 is just a reln which sums up everything dude
Antares
what does e=mc^2 stand for?explain
Piyali
m not dude
jyotirmayee
e for speed of light
jyotirmayee
c for speed of light
jyotirmayee
m for mass of object
jyotirmayee
black hole is a super massive object in space gravitation pull of which s so strong dt evn light cannot escape. John wheeler 1st detected it, n concept of it ws given by Einstein himself from his general theory of relativity. Basically v cn si 3 types of it super massive, interstellar n intermediate
Antares
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jyotirmayee
e defined? sorry it,s wrong
jyotirmayee
how black holes r formed?
Piyali
e is not speed of light
jyotirmayee
black holes formed when the centre of very massive star collapsed itself
jyotirmayee
e is energy.
Antares
what do u mean by quantum mechanics?
jyotirmayee
in most of d cases black holes r formed by massive collapsed star or star system
Antares
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Antares
Apollo is the name of a satellite !!!!!
jyotirmayee
quantum mechanics is d study of physics describing nature at d smallest level of energy of atoms n subatomic particles.
Antares
qm means explains about the microscopic particles..
KRANTHI
nope name of d sun god
Antares
why the mercury used in thermometers?
jyotirmayee
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jyotirmayee
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jyotirmayee
why the mouth became red colour ,,by the regular eating of leaf called ,,betel combining with areca?
jyotirmayee
quantum mechanics is the study of the photon the light particle
Agrim
actually E=mc^2 is only the rest energy of the object and a simplified version of the expansion that covers momentum and objects close to the speed of light
quantum mechanics is the study of sub atomic particles
ofcourse
Arzoodan
quantum mechanics is study of microscope partical
Dharmdev
all formula for calculate specific latent heat of any substance
hello
Nigar
real
Nigar
c'mon guys.. let's talk Physics.
Yoblaze
mass multiplied with latent heat of a substance
Yoblaze
you both?
jyotirmayee
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jyotirmayee
physics is the only subject which underestimated chemistry ,,bio
jyotirmayee
how capacitor is made in inst?
Piyali
what is dipole moment
it is the product of electric charges and distance between the two charges
Shikhar
mishra true thanks dia
Ssempala
product of separation of the poles, the rest shikhar got is right
brad is separation and distance ,are they different?
Ssempala
What is actually a dipole? I know charge separated by a certain distance.... but what does that really mean? what happens in a dipole? why are the charge of same magnitude?
Monalisa
Dipoles forming as a result of the unbalanced distribution of electrons in asymmetrical molecules
Heeran
What is dielectric
its a type of medium. generally poor conductors. but their conductivity can be changed
vedanth
you just have to add impurities
vedanth
Thanks
Ssempala
grt
Ssempala
a material which behave as conductor
Shikhar
insulating material, energy level for electron transfer is very high e.g used to increase a magnetic field in a capacitor
What is the difference between specific heat capacity and heat capacity? Give the equations