# 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:

what is flux
Total number of field lines crossing the surface area
Kamru
Basically flux in general is amount of anything...In Electricity and Magnetism it is the total no..of electric field lines or Magnetic field lines passing normally through the suface
prince
what is temperature change
Celine
a bottle of soft drink was removed from refrigerator and after some time, it was observed that its temperature has increased by 15 degree Celsius, what is the temperature change in degree Fahrenheit and degree Celsius
Celine
process whereby the degree of hotness of a body (or medium) changes
Salim
Q=mcΔT
Salim
where The letter "Q" is the heat transferred in an exchange in calories, "m" is the mass of the substance being heated in grams, "c" is its specific heat capacity and the static value, and "ΔT" is its change in temperature in degrees Celsius to reflect the change in temperature.
Salim
what was the temperature of the soft drink when it was removed ?
Salim
15 degree Celsius
Celine
15 degree
Celine
ok I think is just conversion
Salim
15 degree Celsius to Fahrenheit
Salim
0 degree Celsius = 32 Fahrenheit
Salim
15 degree Celsius = (15×1.8)+32 =59 Fahrenheit
Salim
I dont understand
Celine
the question said you should convert 15 degree Celsius to Fahrenheit
Salim
To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.
Salim
what is d final ans for Fahrenheit and Celsius
Celine
it said what is temperature change in Fahrenheit and Celsius
Celine
the 15 is already in Celsius
Salim
So the final answer for Fahrenheit is 59
Salim
what is d final ans for Fahrenheit and Celsius
Celine
what are the effects of placing a dielectric between the plates of a capacitor
increase the capacitance.
Jorge
besides increasing the capacitance, is there any?
Bundi
mechanical stiffness and small size
Jorge
why for an ideal gas internal energy is directly proportional to thermodynamics temperature?
two charged particles are 8.45cm apart. They are moved and the force on each of them is found to have tripled. How far are they now?
what is flux
Bundi
determining dimensional correctness
determine dimensional correctness of,T=2π√L/g
PATRICK
somebody help me answer the question above
PATRICK
calculate the heat flow per square meter through a mineral roll insulation 5cm thick if the temperature on the two surfaces are 30degree Celsius and 20 degree Celsius respectively. thermal conduction of mineral roll is 0.04
what are the elementary compositions of a cell?
poles, chemical
prabir
when a current pass through a material does the velocity varies
no.
prabir
what is spin entropy ?and disorder in ferromagnetic material
diagram of an hall effect sensor
if a magnetised wire having dipole moment M is bent in the form of arc subtending angle of 45°at centre,new magnetic moment is
prabir
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)
when a current pass through a material does the velocity varies
lovet
yes at adiabatic compression temperature increase
Nepal
how to draw a diagram of a triode
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