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W = V 1 V 2 p d V = V 1 V 2 ( n R T V ) d V .
The figure is a plot of p on the vertical axis as a function of V on the horizontal axis. Two pressures are indicated on the vertical axis, p 1 and p 2, with p 1 greater than p 2. Two volumes are indicated on the horizontal axis, V 1 and V 2, with V 1 less than V 2.  Four points, A, B, C, and D are labeled. Point A is at V 1, p 1. Point B is at V 2, p 1. Point C is at V 2, p 2. Point D is at V 1, p 2. A straight horizontal line connects A to B, with an arrow pointing to the right indicating the direction from A to B. A straight vertical line connects B to C, with an arrow downward indicating the direction from B to C. A straight vertical line connects A to D, with an arrow pointing downward indicating the direction from A to D. A straight horizontal line connects D to C, with an arrow to the right indicating the direction from D to C. Finally, a curved line connects A to C with an arrow pointing in the direction from A to C.
The paths ABC , AC , and ADC represent three different quasi-static transitions between the equilibrium states A and C .

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 = n R T V 1 V 2 d V V = n R T ln 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 to 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 = V 1 V 2 p d V = p 1 V 1 V 2 d V = p 1 ( V 2 V 1 ) .

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 ( V 2 V 1 ) + 0 = p 1 ( V 2 V 1 ) .

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

( p + a V 2 ) ( V b ) = R T ,

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?


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.


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

p = R T V b 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

W = V 1 V 2 ( R T V b a V 2 ) = | R T ln ( V b ) + a V | V 1 V 2 = R T ln ( V 2 b V 1 b ) + a ( 1 V 2 1 V 1 ) .


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.

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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 ( V 2 V 1 )

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Internal energy

The internal energy     E 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:

Questions & Answers

Using Kirchhoff's rules, when choosing your loops, can you choose a loop that doesn't have a voltage?
Michael Reply
how was the check your understand 12.7 solved?
Bysteria Reply
LOAK Reply
he's the father of 3 newton law
he is Chris Issaac's father :)
how to name covalent bond
Bryan Reply
what do you understand by the drift voltage
Brunelle Reply
what do you understand by drift velocity
well when you apply a small electric field to a conductor that causes to add a little velocity to charged particle than usual, which become their average speed, that is what we call a drift.
drift velocity
what is an electromotive force?
Danilo Reply
It is the amount of other forms of energy converted into electrical energy per unit charge that flow through it.
How electromotive force is differentiated from the terminal voltage?
in the emf power is generated while in the terminal pd power is lost.
what is then chemical name of NaCl
Sagar Reply
sodium chloride
sodium chloride
Sodium Chloride.
How can we differentiate between static point and test charge?
Comfort Reply
Wat is coplanar in physics
Humble Reply
two point charges +30c and +10c are separated by a distance of 80cm,compute the electric intensity and force on a +5×10^-6c charge place midway between the charges
Tijani Reply
what is the difference between temperature and heat
Ishom Reply
Heat is the condition or quality of being hot While Temperature is ameasure of cold or heat, often measurable with a thermometer
Temperature is the one of heat indicators of materials that can be measured with thermometers, and Heat is the quantity of calor content in material that can be measured with calorimetry.
the average kinetic energy of molecules is called temperature. heat is the method or mode to transfer energy to molecules of an object but randomly, while work is the method to transfer energy to molecules in such manner that every molecules get moved in one direction.
2. A brass rod of length 50cm and diameter 3mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°c( degree Celsius) if the original length are 40°c(degree Celsius) is there at thermal stress developed at the junction? The end of the rod are free to expand (coefficient of linear expansion of brass = 2.0×10^-5, steel=1.2×10^-5k^1)
A charge insulator can be discharged by passing it just above a flame. Explain.
Mudassar Reply
of the three vectors in the equation F=qv×b which pairs are always at right angles?
what is an ideal gas?
Justine Reply
What is meant by zero Kelvin ?
Why does water cool when put in the pot ?
when we pour the water in a vessel(pot) the hot body(water) loses its heat to the surrounding in order to maintain thermal equilibrium.Thus,water cools.
when we drop water in the pot, the pot body loses heat to surrounded in order to maintain thermal equilibrium thus,water cool.
my personal opinion ideal gas means doesn't exist any gas that obey all rules that is made for gases, like when get the temp of a gas lower, it's volume decreases.since the gas will convert to liquid when the temp get lowest.. so you can imagine it, but you can't get a gas at the lowest T
Edit An ideal gas is a theoretically gascomposed of many randomly moving point particles whose only interactions are perfectly elastic collisions.
ideal gases are real gases at low temperature
Practice Key Terms 2

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