Kinetic theory of gasses Integration problem

In summary: To see how it works, let's do it for the simplest case, no T^2. ThenV = (24.9 J/K) T/PdV = (24.9 J/K)/P dTsoW = integral((24.9 J/K)/P dT, T1, T2)= (24.9 J/K)/P (T2 - T1)= P V [(T2 - T1)/V]= P (V2 - V1)
  • #1
brett812718
57
0

Homework Statement


In the temperature range 310 K and 330 K, the pressure p of a certain nonideal gas is related to volume V and temperature T by
p = (24.9 J/K) T/V - (0.00662 J/K2)T^2/V
How much work is done by the gas if its temperature is raised from 314 K to 324 K while the pressure is held constant?


Homework Equations


Pv=NRT
W=integral(p dv)


The Attempt at a Solution



W=integral((24.9 J/K) T/V - (0.00662 J/K2)T^2/V dv)
I am not sure what to use for the limits of integration. the problem gives me a range of temperatures, is there any way that I can use those? I am very sure the limits of integration should be in units of m^3.
 
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  • #2
HINT: Can you use the given equation to calculate the change in volume from the change in temperature?
 
  • #3
Then I get V1=7165.9J/P and V2=7414.6J/P. can I use those as the limits of Integration?
 
  • #4
brett812718 said:
Then I get V1=7165.9J/P and V2=7414.6J/P. can I use those as the limits of Integration?
I would say so :smile:
 
  • #5
what do I do with the T and T^2 parts of the pressure equation?
 
  • #6
brett812718 said:
what do I do with the T and T^2 parts of the pressure equation?
Notice that when you substitute the limits in, the expressions for pressure cancel.
 
  • #7
I think i am doing somthing wrong. I integrate the equation p = (24.9 J/K) T/V - (0.00662 J/K2)T^2/V right? so what do I do with the T and T^2? Would I treat them as a constant during integration?
 
  • #8
No, T is not constant. You know that
p = (24.9 J/K) T/V - (0.00662 J/K2)T^2/V
with P constant. That is the same as
V = (24.9 J/K) T/P - (0.00662 J/K2)T^2/P

dV= [(24.9 J/K)/V - (0.00662 J/K2)(2T)/V]dT

and you can do the entire integral in terms of T.
 
  • #9
how did you get p to cancel out?
 
  • #10
brett812718 said:
how did you get p to cancel out?
'Cancel out' was a poor phrase to use. I should have said 'change of variable', it's been a long day :zzz:
 

What is the Kinetic Theory of Gases?

The Kinetic Theory of Gases is a scientific model used to explain the behavior of gases at a molecular level. It states that gases are composed of particles that are constantly in random motion and that the temperature of a gas is directly related to the average kinetic energy of its particles.

How does the Kinetic Theory of Gases explain the properties of gases?

The Kinetic Theory of Gases explains the properties of gases by stating that the particles in a gas have no definite shape or volume and are constantly in motion. It also explains how gases can be compressed, have low densities, and can diffuse easily due to the constant motion of its particles.

What is an integration problem in the context of the Kinetic Theory of Gases?

An integration problem in the context of the Kinetic Theory of Gases refers to solving mathematical equations or formulas that involve integrating the velocity distribution of gas particles over a given range. This is often used to calculate the average kinetic energy of gas particles or other properties of gases.

What are some applications of the Kinetic Theory of Gases?

The Kinetic Theory of Gases has many practical applications, including predicting the behavior of gases in various environments, such as in engines, refrigerators, and weather systems. It is also used in the development of new materials and technologies, such as in the design of solar cells and nanomaterials.

Are there any limitations to the Kinetic Theory of Gases?

While the Kinetic Theory of Gases is a useful model for understanding the behavior of gases, it does have some limitations. For example, it assumes that gas particles have no interactions with each other, which is not always the case. It also does not account for quantum effects, which are important in certain situations, such as at very low temperatures.

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