How much work is done by inflating a balloon?

[herbally]

Homework Statement

An initially deflated and flat balloon is connected by a valve to a storage tank containing helium gas at 1 MPa at ambient temperature of 20°C. The valve is opened and the balloon is inflated at constant pressure of 100 kPa (atmospheric pressure) until it becomes spherical at D₁ = 1 m. If the balloon is larger than this, the balloon material is stretched giving a pressure inside as:

P = P₀ + C (1-D₁/D) D₁/D

The balloon is slowly inflated to a final diameter of 4 m, at which point the pressure inside is 400 kPa. The temperature remains constant at 20°C. Determine the work done during the overall process.

Homework Equations

W = ∫PdV
PV = nRT

The Attempt at a Solution

I solved for the volumes of the balloon at D₁ = 1 m and at D₁ = 4 m to use for the definite integral. Then I the formula for P in terms of V using the ideal gas law:

P = nRT/V

At this point, I realize that I don't know the number of moles in the balloon and that I haven't used the expression given in the problem for P. I know how to evaluate the calculus, but I'm terrible at setting up the integrals. Anyone have a hint for me? Thanks!

 

[Simon Bridge]

Since the number of moles changes, the state of the gas in the balloon is given by four numbers, (P,V,T,n) related by the ideal gas equation. You have an additional equation to tell you a final P in terms of an initial one. You also know how to get the volumes from the diameters.

The balloon moves between two states where you know the something about the state variables.

As usual, list what you know.
Start with the initial state:
P0=
V0=
T0=
n0=

put a question mark next to anything you don't know, then look for an equation that relates an unknown to the other stuff you do know.

 

[herbally]

Temperature is constant at 20°C.

P₀ = ?
V₀ = .5236 M³
n₀ = 21.49 mol

P_f = 400kPa
V_f = 33.51 m³
n_f = 5502 mol

My inclination is that I should back-solve for P₀ using the given expression, but I don't know what C is. Heat capacity is my only thought, but neither pressure or volume is constant. Am I even close?

 

[Simon Bridge]

What's wrong with using the ideal gas equation to get P0?
Hang on - aren't you told the initial pressure is 100kPa??
How did you get the number of moles?

What you need is the equation for P as a function of volume.
This means you need a value for C - you can determine that from the information you have.
(Check your idea about C by doing a dimensional analysis - what units should it have?)

 

[herbally]

Not real sure how to interpret the problem there. I know the initial pressure in the tank is 1MPa and that the pressure added to the balloon is 100kPa, but I assume that P₀ is the pressure when the balloon becomes spherical. Am I misinterpreting?

EDIT: Just realized that I calculated n₀ using 100kPa as P₀.

That said, it can't be right because filling the balloon at that pressure doesn't mean that the balloon is spherical at 100kPa.

 

[herbally]

From the given expression, C should be in Pascals.

 

[Simon Bridge]

The problem statement says the balloon is spherical at 100kPa - which it calls "atmospherc pressure".
The clue is that the balloon if filled at that pressure until it stops getting bigger, and it has not stretched yet.

This means you have a complete description of the initial and final states of the air in the balloon.
From that you can compute C.

From the given expression, C should be in Pascals.

... i.e. C should have units of pressure.
What are the units of "heat capacity"?

The work done on the gas is the area under the P-V graph.
This is where the P function comes in.

 

[herbally]

Heat capacity has units of J/g*K.

I plugged 100 kPa in for P₀, and 400 kPa for P and solved for C yielding a value of 1600 kPa.

So then I plugged those values into the given expression and simplified yielding:

P = 100 kPa + 1600 kPa/D - 1600 kPa/D

So I guess I need this expression in terms of volume to solve for work?

 

[Simon Bridge]

P = 100 kPa + 1600 kPa/D - 1600 kPa/D

It is important to think about what you write down.
Does this equation make sense? Don't the last two terms cancel out?

 

[herbally]

The last term should have been 1600/D².

Sorry.

 

[Simon Bridge]

OK:

P = 100 kPa + 1600 kPa/D - 1600 kPa/D^2

... does this make sense?
If the diameter were D=6m, would you expect the pressure to make that happen to be bigger or smaller than the pressure for D=4m?
Now what does the equation tell you?

 

[herbally]

You're right. Doesn't make much sense since the pressure would get smaller as diameter increases.

Back to the drawing board.

 

[herbally]

Kind of stuck. Not sure where to go from here.

 

[Simon Bridge]

Well, in the problem statement, initially D₁=100kPa and finally D₁=400kPa ... what does that suggest to you?

 

[herbally]

I realize that pressure and volume are changing. I assume that I need an expression for pressure in terms of volume. Since the given expression for pressure is in terms of diameter, I solved the equation for volume of a sphere for diameter, but, plugging that into the given expression for pressure yields an expression that results in a far more complex integration which suggests to me that something isn't right. (Most of the solutions to the problems in this book are conceptual challenges with simpler computations.)

 

[Simon Bridge]

There are two diameters in your equation - which is the diameter that the problem statement says is changing?

 

[herbally]

Not sure but tend to think it's D₁.

 

[Simon Bridge]

So what happens if you redo your derivation using D1 as the variable?

 

[herbally]

When I integrate the expression for pressure in terms of volume, I get an answer that is close to the correct answer but still off by a few thousand joules.

I think my expression is wrong. I took the equation for volume of a sphere and solved it for diameter. Then I took that solution and plugged it into the given expression for D₁. Is that not right?

 

[Simon Bridge]

Well that's what I would have done - after getting the correct value for C.