Volume Expansion: Answers to Homework Questions

[negation]

Homework Statement

A standard mercury thermometer consists of a hollow glass cylinder, the stem, attached to a bulb filled with mercury. As the temperature of the thermometer changes, the mercury expands (or contracts) and the height of the mercury column in the stem changes. Marks are made on the stem to denote the height of the mercury column at different temperatures such as the freezing point (0∘C) and the boiling point (100∘C) of water. Other temperature markings are interpolated between these two points.

Due to concerns about the toxic properties of mercury, many thermometers are made with other liquids. Consider draining the mercury from the above thermometer and replacing it with another, such as alcohol. Alcohol has a coefficient of volume expansion 5.6 times greater than that of mercury. The amount of alcohol is adjusted such that when placed in ice water, the thermometer accurately records 0∘C. No other changes are made to the thermometer.

1) When the alcohol thermometer is placed in 20∘Cwater, what temperature will the thermometer record?

2)
When the alcohol thermometer is placed in a −10∘C substance, what temperature will the thermometer record?3)If you want to design a thermometer with the same spacing between temperature markings as a mercury thermometer, how must the diameter of the inner hollow cylinder of the stem of the alcohol thermometer compare to that of the mercury thermometer? Assume that the bulb has a much larger volume than the stem.(solved)

The Attempt at a Solution

ΔV_A = V_AiαΔT

The cylinder has a volume: πr²h

Change in volume of alcohol in a cylinder: ΔV_A = πr²h(5.6α)ΔT

Change of volume of mercury in a cylinder: ΔV_M = πr²h.α.ΔT

I am experiencing a confusion here. Question states that the thermometer is placed into a water of 20°C. However, nothing was said about the initial temperature. Without this piece of information I cannot utilize ΔT.

 

[BvU]

You know the reading at 0 degrees C. So 20^∘ would be a good ΔT. You also know that the corresponding ΔV_M makes the mercury rise to the 20[sup∘[/sup]C mark.

 

[negation]

You know the reading at 0 degrees C. So 20^∘ would be a good ΔT. You also know that the corresponding ΔV_M makes the mercury rise to the 20[sup∘[/sup]C mark.

I don't...
what is the height of the thermometer at 0 and 20 degree celsius?

 

[BvU]

Give it a name: "0". The makers of the device make a mark and write a zero at that height. Then the Hg is drained and the C₂H₅OH is filled up to that height on the scale.

[edit] as for the 20 one: that's the real exercise for you to do. It has to be calculated from the difference in ΔV.

 

[HalfLostAndSearching]

1) Without knowing the initial temperature of the alcohol thermometer, we cannot accurately determine the temperature it will record when placed in 20°C water. However, we can make some assumptions and calculations based on the given information.

Assuming the initial temperature of the alcohol thermometer is 0°C, the change in temperature would be 20°C. Using the formula ΔVA = VAiαΔT, we can calculate the change in volume of the alcohol in the thermometer.

ΔVA = VAiαΔT
ΔVA = VAi * (5.6α) * 20°C

Now, we know that the amount of alcohol is adjusted such that when placed in ice water, the thermometer accurately records 0°C. This means that the volume of alcohol at 0°C must be the same as the volume of mercury at 0°C in the original thermometer. Therefore, we can equate the initial volume of the alcohol with the initial volume of the mercury.

VAi = VMi

Substituting this into our equation, we get:

ΔVA = VMi * (5.6α) * 20°C

Now we can compare this change in volume to the change in volume of mercury at 20°C in the original thermometer.

ΔVA = ΔVM

VMi * (5.6α) * 20°C = VMi * α * 20°C

Simplifying, we get:

5.6α = α

This means that the coefficient of volume expansion for alcohol must be equal to the coefficient of volume expansion for mercury. Therefore, if the initial volume of alcohol is the same as the initial volume of mercury, the temperature recorded by the alcohol thermometer when placed in 20°C water would be the same as the temperature recorded by the original mercury thermometer.

2) Similarly, without knowing the initial temperature of the alcohol thermometer, we cannot accurately determine the temperature it will record when placed in a -10°C substance. However, using the same assumptions and calculations as in part 1, we can determine that the temperature recorded by the alcohol thermometer would be the same as the temperature recorded by the original mercury thermometer.

3) To design a thermometer with the same spacing between temperature markings as a mercury thermometer, the diameter of the inner hollow cylinder of the stem of the alcohol thermometer must be smaller than that of the mercury thermometer. This is because the coefficient of volume expansion for alcohol is 5.6