Determination Of Coefficient Of Linear Expansion Of A Metal Rod 6a5o2h

Laboratory report DETERMINATION OF COEFFICIENT OF LINEAR EXPANSION OF A METAL ROD

Objective: The purpose of this experiment is to observe linear expansion of a metal rod and to determine the value of the coefficient of thermal expansion of a metal. Theory: Thermal expansion is the tendency of matter to change in shape, area, and volume in response to a change in temperature, through heat transfer. (This experiment touches only the linear (1 dimension) part of thermal expansion). Atoms and molecules in a solid constantly oscillate around its equilibrium point. This kind of excitation is called thermal motion. When a substance is heated, its constituent particles begin moving more, thus maintaining a greater average separation with their neighboring particles. Unlike gases or liquids, solid materials tend to keep their shape when undergoing thermal expansion. Thermal expansion generally decreases with increasing bond energy, which also has an effect on the melting point of solids, so, high melting point materials are more likely to have lower thermal expansion. The degree of expansion divided by the change in temperature is called the material's coefficient of thermal expansion and generally varies with temperature. It is the fractional change in length per degree of temperature change. ∝L =

1 dL L dT

where L is a particular length measurement and

dL dT

is the rate of change of that linear dimension

per unit change in temperature. The change in the linear dimension can be estimated to be: ∆L =∝L ∆ T L

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This equation works well as long as the linear-expansion coefficient does not change much over the ∆L change in temperature ∆ T , and the fractional change in length is small L ≪1 . If either of these conditions does not hold, the equation must be integrated.

Procedure: In order to get reliable data an apparatus of linear expansion was used. The apparatus consists of a hollow metal rod (1) (placed between two frames, which one of them is fixed to the rod) fitted with a heating spiral (2). The power is changed by the controller (3) which has 10 positions (1 supplying minimal power and 10 maximal power). The temperature is measured with the thermoelement (4). The milivoltmeter (5) together with the graduation graph Δt = f (U mV) are used to find the change of the temperature Δt = t – t0 respect of the room temperature t0. The thermal expansion of the metal rod is measured with indicator (6) which has two scales (the small one shows millimeters and the big one show the hundredth of millimeters). The indicator is fitted on the non-fixed end of the rod. Equipment: 1 – rod; 2 – heating spiral; 3 – power controller; 4 – thermoelement; 5 –milivoltmeter; 6 – indicator

Sequence of the experiment: 1. The length L0 of the metal rod was measured using a ruler (from frame to frame). The room temperature t0 was also measured using a wall thermometer. 2. The scales of the indicator were calibrated. The small scale was set to 4 mm mark and the big scale was set to 0 mark. 3. The apparatus control switch was set to position 1 and connected to the electric power source. 4. The apparatus was switched on. 5. Wait till the milivoltmeters readings stabilized (can take up to 10min). 6. After it had stabilized, the data was collected and the apparatus switched to position 2. 7. The process was repeated until the position 10 was reached and all the data collected. 8. The data was processed and analyzed.

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Results: t0 = 21.2 oC The voltage of Temperature the change* thermoelement, Δti = t – t0 , mV °C

No .

1 2 3 4 5 6 7** 8** 9** 10

5.1 6.05 6.2 6.75 7.2 7.4 9

70 82 83 94 96 102 124

n0 = 4 mm Indicator Absolute value of reading thermal ni , expansion mm li = |ni – n0|, mm 3.36 0.64 3.26 0.74 3.246 0.755 3.19 0.81 3.13 0.87 3.1 0.9 2.92 1.02

L0 = 0.46 m = 460 mm

* The values were taken from the graduation graph Δt = f (UmV), included with the apparatus ** Because of shortage of time, the apparatus positions 7,8,9 were skipped Calculations of the data table: 

Absolute value of thermal expansion li = |ni – n0| 1. l1 = |n1 – n0| = |3.36 - 4| = 0.64 (mm) 2. l2 = |n2 – n0| = |3.26 - 4| = 0.74 (mm) 3. l3 = |n3 – n0| = |3.246 - 4| = 0.755 (mm) 4. l4 = |n4 – n0| = |3.19 - 4| = 0.81 (mm) 5. l5 = |n5 – n0| = |3.13 - 4| = 0.87 (mm) 6. l6 = |n6 – n0| = |3.1 - 4| = 0.9 (mm) 7. l7 = |n7 – n0| 8. l8 = |n8 – n0| 9. l9 = |n9 – n0| 10. l10 = |n10 – n0| = |2.92 - 4| = 1.02 (mm)

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Thermal expansion versus temperature change l = f (t - t0) 1.2 1

0.76 0.640.74

0.8

l, mm

0.9 0.87 0.87 0.81

1.02

0.6 0.4 0.2 0

0

20

40

60

80

100

120

140

t - t0, 0C

Because the positions of collected data are scattered and not located in a straight line the trendline was drawn. In order to calculate the coefficient of linear expansion a random value (T – t0, ∆ L) from the trendline must be taken. And to check the validity of the calculation an exacted value (t6-t0, l6) from the table was taken. ∝v =

l L0 (t −t 0 )

Trendline value (T – t0, ∆

I. ∝v =

L)

∆L L0 (T −t 0 )

∆ L = 0.868 mm T – t0 = 100 0C ∝v =

II.

0.868 mm 0.000868 m 217 = = ℃−1 ≈ 1.887 ×10−5 ℃−1 460 mm ×100 ℃ 0.46 m× 100℃ 11500000

Table value for comparison (t6-t0, l6)

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∝6 =

l6 L0(t 6 −t 0 )

l6 = 0.9 mm t6 – t0 = 102 0C ∝6 =

0.9 mm 0.0009 m 3 = = ℃−1 ≈ 1.918 ×10−5 ℃−1 460 mm ×102 ℃ 0.46 m×102 ℃ 156400

Conclusion: The experiment in its entirety went well. The data and values all fall in an expected range. When the apparatus was activated, the metal rod (which initial length was 0.46 meters) gradually expanded for about 1 mm in its length and reached approximately 145 ℃ temperature. −5 −1 The calculated average coefficient of linear expansion of a metal rod was ∝v =1.887 ×10 ℃ . In −5 −1 addition to compare and check logical validity of the ∝v value, ∝6=1.918 ×10 ℃ was

calculated. As seen the difference is rather small for about 1.6 % between ∝v and ∝6 . The factors that probably had negative effect on the precision of the experiment were: the outdated apparatus, time shortage and temperature determination by a given graph rather than a thermometer. To improve the laboratory work: the experiment should be made with different kinds of metal rods (copper, aluminum, … etc.); thermally isolate the metal rod from the environment; use more precise indicators and temperature measuring apparatus; try other heating types. References: Boundless. “Linear Expansion.” Boundless Physics. Boundless, 26 May. 2016. Retrieved 11 Sep. 2016 from https://www.boundless.com/physics/textbooks/boundless-physicstextbook/temperature-and-kinetic-theory-12/thermal-expansion-103/linear-expansion-370-5647/ https://en.wikipedia.org/wiki/Thermal_expansion https://www.youtube.com/watch?v=m7tTUW1XHbQ https://www.youtube.com/watch?v=dmq_X5vtPoA https://www.youtube.com/watch?v=KyaM7O6E3xM 

Laboratory work “DETERMINATION OF COEFFICIENT OF LINEAR EXPANSION OF A METAL ROD”

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