Solving Volumetric Expansion of Water in Glass

[chopnhack]

Homework Statement

An ordinary glass is filled to the brim with water at 100oc. How much water could be added to the glass if the temperature is lowered to 20oC? Assume that the coefficient of volume expansion for glass is 2.7 x 10-5 K-1 and for water it is 2.1 x 10-4 K-1.

Homework Equations

ΔV = γ⋅VΔθ

The Attempt at a Solution

Δθ = 80 K
I will assume 1m³ for volume to make the calculations simple for both.

ΔV_glass = 2.7 x 10-5 K-1⋅1m³⋅80k = 0.00216m³

ΔV_water = 2.1 x 10-4 K-1 ⋅ 1m³⋅80k = 0.0168m³

I reasoned that since there is differential shrinkage, the glass shrinks 0.00216m³, thus there is less available room for storing water, but the water shrank also, if we subtracted the results from each other, we would have the net change. So we could add 0.01464m³ more water.

I wanted to check my supposition that I could use 1m³ for both the glass and the water. The glass does not occupy a full cubic meter, since its mostly walls with a void in side. Is that right? Or does that not get addressed in intro physics? Have I worked out the problem correctly or ? I probably should report this in percentage as 1.46% change, no?

Thanks!

 

[SammyS]

Homework Statement

An ordinary glass is filled to the brim with water at 100oc. How much water could be added to the glass if the temperature is lowered to 20oC? Assume that the coefficient of volume expansion for glass is 2.7 x 10-5 K-1 and for water it is 2.1 x 10-4 K-1.

Homework Equations

ΔV = γ⋅VΔθ

The Attempt at a Solution

Δθ = 80 K
I will assume 1m³ for volume to make the calculations simple for both.

ΔV_glass = 2.7 x 10-5 K-1⋅1m³⋅80k = 0.00216m³

ΔV_water = 2.1 x 10-4 K-1 ⋅ 1m³⋅80k = 0.0168m³

I reasoned that since there is differential shrinkage, the glass shrinks 0.00216m³, thus there is less available room for storing water, but the water shrank also, if we subtracted the results from each other, we would have the net change. So we could add 0.01464m³ more water.

I wanted to check my supposition that I could use 1m³ for both the glass and the water. The glass does not occupy a full cubic meter, since its mostly walls with a void in side. Is that right? Or does that not get addressed in intro physics? Have I worked out the problem correctly or ? I probably should report this in percentage as 1.46% change, no?

Thanks!

Hint: Suppose you replace the water in the full glass with an equivalent volume of glass. Won't the glass remain full, no matter the temperature ?

 

[chopnhack]

If this was all glass, then the entire piece would expand and contract together.

What were you hinting at? Did I get it wrong?

 

[SammyS]

If this was all glass, then the entire piece would expand and contract together.

What were you hinting at? Did I get it wrong?

You seemed rather uncertain as to whether your supposition was correct so I tried to give a hint to help your confidence.

I wanted to check my supposition that I could use 1m³ for both the glass and the water. The glass does not occupy a full cubic meter, since its mostly walls with a void in side. Is that right? Or does that not get addressed in intro physics? Have I worked out the problem correctly or ? I probably should report this in percentage as 1.46% change, no?

You could have used something more general for the volume, such as V₀, or V₁₀₀. The main point, however, is to use this for both glass and water.

By the way:
In the blue bar above the window for composing a post, find X² and X₂ for superscript and subscript. Also find there ∑ to give a large set of symbols, including the degree ° symbol.

Even better is to learn a little LaTeX .

Added in Edit: Your result looks correct to me.

 

[chopnhack]

Gotcha, thank you for confirming the work. I hope you can understand my trepidation - the glass itself doesn't occupy a cubic meter, so when I made that assumption, I was concerned that I had missed something fundamental. I had copied and pasted the question and didn't go back to revise it. I am familiar with the site tools now, but latex still gives me a rash ;-)
cheers

 

[haruspex]

I probably should report this in percentage as 1.46% change

As I read the question, it is asking for an actual volume. (If you quote a percentage, would that be of the original volume or of the new volume?)

 

[chopnhack]

As I read the question, it is asking for an actual volume. (If you quote a percentage, would that be of the original volume or of the new volume?)

I wrote it up as the ability to add ~1.4% more water. I would take that as an increase over the original volume.