- #1

KingDaniel

- 44

- 1

ie: How do you derive the pV Work formula, W= "integral"dW= "integral"pdV

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- Thread starter KingDaniel
- Start date

In summary, the conversation discusses deriving the work done in changing from State 1 to State 2 in Thermodynamics. The participants discuss the formula W = ∫F.dx and how it can be written as W = ∫PdV, with P being the pressure and A being the area. They also discuss the possibility of F and P not being constant, and how the integral can still be written without them. Finally, they clarify the concept of using A(x)dx as a small volume and how it relates to the work equation.

- #1

KingDaniel

- 44

- 1

ie: How do you derive the pV Work formula, W= "integral"dW= "integral"pdV

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- #2

calculo2718

- 25

- 1

We can substitute F in the in the work equation (assuming constant force) and obtain [;W = \int PA dx;] which is effectively [;W = \int PdV;].

- #3

KingDaniel

- 44

- 1

W = PA * \int dx , since PA is constant,

and finally W = PAx , since "\int dx"=x,

and not W = \int PdV

I thought that maybe Ax could be considered as dV, but I'm not sure.

Please try to explain further as there might be something I'm not considering

- #4

RedDelicious

- 138

- 43

- #5

Qwertywerty

- 501

- 66

It would , however , be a constant ( approximately ) over a length dx . So we write F as P.A at and write the work as ( ∫PA.dx ) .

A.dx is the small volume dV that is crossed by an acting force . So we write the work done as ( ∫P.dV ) ,where P is not necessarily a constant .

- #6

KingDaniel

- 44

- 1

@Qwertywerty , what you've said seems to makes sense and I want to understand it this way but I need to clear a few things first:

If you integrate F with respect to x, you get F*x (which is what we were taught in high school physics, without the integral). So if you integrate, pA with respect to x, shouldn't you get pAx = pV, and not integral of pdV? If F is not constant, don't we just treat it as average of all the F's, and still treat it as a constant?

- #7

calculo2718

- 25

- 1

KingDaniel said:

@Qwertywerty , what you've said seems to makes sense and I want to understand it this way but I need to clear a few things first:

If you integrate F with respect to x, you get F*x (which is what we were taught in high school physics, without the integral). So if you integrate, pA with respect to x, shouldn't you get pAx = pV, and not integral of pdV? If F is not constant, don't we just treat it as average of all the F's, and still treat it as a constant?

Yes, you learned W = Fx in high school but that was assuming F was constant, I shouldn't have said assuming F is constant, but in my defense it was nearly 5 in the morning when I posted.

Think about it like this, force is a function of position(not necessarily constant), area is also a function of position so its more like W = [;\int F(x)dx;] and [;W = \int P(x)A(x)dx;], [;A(x)dx;] is essentially [;dV;].

- #8

Qwertywerty

- 501

- 66

∫kx.dx = k(x∧2)/2 + c

∫k(x∧2).dx = k(x∧3)/3 + c

and so on , where k and c are some random constants .

Now suppose F = kx , could you remove F from the integral of ∫F.dx ?

F = P.A will however be true at any instant .

Thus we can write W =∫F.dx or W= ∫PA.dx .

In the second case it is not necessary that P or A is a constant over some large distance . They may simply be functions as P(x) and A(x) . So can you now remove either of them from your integral ?

However we can write A(x).dx as the small volume that that the particle , on which the force acts , moves through at any x .

Thus work can be written as W = ∫P.dV .

I hope this helps .

The pV Work formula, also known as the integral work formula, is derived from the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. By rearranging this equation and substituting the definition of work (W = Fd), we can arrive at the pV Work formula, W = integral pdV.

The "integral" in the pV Work formula represents the mathematical concept of integration, which is a way to calculate the total area under a curve. In this formula, the integral is used to calculate the work done by the system as the pressure changes over a volume range.

The pV Work formula is used in thermodynamics to calculate the work done by a system as it undergoes a change in volume. This can be useful in analyzing the efficiency of an engine or in determining the energy required to compress a gas.

The units of the pV Work formula are Joules (J), as work is measured in energy units. The pressure (p) is typically measured in Pascals (Pa) and the volume (V) in cubic meters (m^3), resulting in the units J = Pa*m^3.

Yes, the pV Work formula assumes that the pressure remains constant throughout the volume change, and that the work done is reversible (meaning that the system can be returned to its original state without any energy loss). Additionally, the formula does not take into account any non-pV work, such as electrical or magnetic work, that may be present in a system.

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