Mathematical interpretation of work done by a gas

In summary: Thanks for the advice!In summary, the person is asking about pressure and how it is related to work. They read in a different source that work can be expressed as an integral when pressure varies, so they need to evaluate that.
  • #1
MexChemE
237
55
Hello everybody! First of all, I would like to say that this is my first post in this forum, even though I've occasionally read some posts before. I'm a ChemE major from Mexico!

I am currently taking Thermodynamics I, and I have trouble figuring out the expression: [tex]W = \int_{V_1}^{V_2} PdV[/tex] I suppose pressure is being treated like a constant in the above equation. If so, if we integrate we would get this expression: [itex]W = P(V_2-V_1)[/itex], right? But then I read in HyperPhysics that we express work as an integral when pressure varies too, so, wouldn't we need an integral of two variables?

I know we express work as an integral because it is the area under the PV curve, but as you can see I'm currently lost between concepts. I hope my question makes sense. Thanks in advance!
 
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  • #2
MexChemE said:
I suppose pressure is being treated like a constant in the above equation.
No. In general, the pressure could be a function of volume.

If so, if we integrate we would get this expression: [itex]W = P(V_2-V_1)[/itex], right?
Right. For an isobaric process.

But then I read in HyperPhysics that we express work as an integral when pressure varies too, so, wouldn't we need an integral of two variables?
You'd need to evaluate:
[tex]W = \int_{V_1}^{V_2} P(V)dV[/tex]
 
  • #3
P is the function that is being integrated. V is the variable of integration.
 
  • #4
I never considered pressure as a function, that was the problem. It makes sense now. *sigh* Thank you both!
 
  • #5
MexChemE said:
I never considered pressure as a function, that was the problem. It makes sense now. *sigh* Thank you both!
Hi MexChemE. Welcome to Physics Forums! I too am a ChE.

For more on this subject, see my Blog on my PF personal page. I treat a lot of issues that people who are new to Thermodynamics get confused about. It is only a couple of pages long.

Chet
 
  • #6
Thanks Chet! I am definitely going to check your blog, thermodynamics is actually my favorite branch of physics. Here in Mexico, we have the option of presenting an undergraduate research thesis in order to obtain our degree, and I'm planning on focusing my research on thermodynamics. I'm just a freshman currently, though.
 

FAQ: Mathematical interpretation of work done by a gas

1. What is the mathematical definition of work done by a gas?

The mathematical definition of work done by a gas is the product of the force applied to the gas and the distance over which the force is applied. It is represented by the equation W = F x d, where W is work, F is force, and d is distance.

2. How is the work done by a gas related to its pressure and volume?

The work done by a gas is directly proportional to its pressure and volume. This means that as the pressure or volume of the gas increases, the work done by the gas also increases. This relationship is described by the equation W = P x ΔV, where W is work, P is pressure, and ΔV is the change in volume.

3. What is the difference between positive and negative work done by a gas?

Positive work done by a gas occurs when the gas expands, meaning that the force and displacement are in the same direction. Negative work is done when the gas is compressed, meaning that the force and displacement are in opposite directions. This can be represented by the sign of the work, with positive work being denoted by a plus sign and negative work by a minus sign.

4. How is the work done by a gas related to its internal energy?

The work done by a gas is equal to the change in its internal energy. This means that when work is done on a gas, its internal energy increases, and when work is done by a gas, its internal energy decreases. This relationship is described by the equation W = ΔU, where W is work and ΔU is the change in internal energy.

5. Can the work done by a gas be negative?

Yes, the work done by a gas can be negative. This occurs when the gas is compressed, as explained in question 3. Negative work done by a gas means that the gas is losing energy, which can be seen as a decrease in its internal energy. This is commonly observed in refrigerators and air conditioners, where the gas is compressed to extract heat from the system.

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