Thermal expansion

1. Characterize the thermal expansion of solids, liquids, and gases!

Solution:

Thermal expansion is a change in an object's dimensions caused by a change in its temperature.

A.) Solids

1.) Linear expansion:

l=l0(1+α⋅Δt)l = l_0 (1 + \alpha \cdot \Delta t)

α\alpha = coefficient of thermal linear expansion

  • α(Cu)=1.7×10−5 K−1\alpha(\text{Cu}) = 1.7 \times 10^{-5} \, K^{-1}

  • α(Al)=2.4×10−5 K−1\alpha(\text{Al}) = 2.4 \times 10^{-5} \, K^{-1}

  • α(Fe)=1.2×10−5 K−1\alpha(\text{Fe}) = 1.2 \times 10^{-5} \, K^{-1}

Relative elongation:

ε=Δll0⋅100%\varepsilon = \frac{\Delta l}{l_0} \cdot 100 \%

2.) Volumetric expansion:

V=V0(1+3α⋅Δt)V = V_0 (1 + 3 \alpha \cdot \Delta t) ρ=ρ0(1−3α⋅Δt)\rho = \rho_0 (1 - 3 \alpha \cdot \Delta t)

B.) Liquids

Volumetric expansion:

V=V0(1+β⋅Δt)V = V_0 (1 + \beta \cdot \Delta t)

β\beta = coefficient of thermal volumetric expansion

  • β(H2O)=1.8×10−4 K−1\beta(\text{H}_2\text{O}) = 1.8 \times 10^{-4} \, K^{-1}

  • β(diesel)=1.1×10−3 K−1\beta(\text{diesel}) = 1.1 \times 10^{-3} \, K^{-1}

ρ=ρ0(1−β⋅Δt)\rho = \rho_0 (1 - \beta \cdot \Delta t)

C.) Gases

1.) Volumetric expansion:

V=V0(1+γ⋅Δt)V = V_0 (1 + \gamma \cdot \Delta t)

γ\gamma = coefficient of expansion is the same for all gases:

γ=1273.16 K=0.00366 K−1=3.66×10−3 K−1\gamma = \frac{1}{273.16 \, K} = 0.00366 \, K^{-1} = 3.66 \times 10^{-3} \, K^{-1}


2.A copper wire (α = 1,7.10-5K-1) had a length of 21,55 m at a temperature of -50C. What is its length at 300C? By how many cm did the wire lengthen?

Solution:

Analysis:

l0 = 21,55 m,  α = 1,7.10-5K-1 = 0,000017 K-1,  Δt = 300C - (-50C) = 350C

fyzika-tepelna-roztaznost-2.gif 

After heating, the wire has a length of 21,56 m. It lengthened by 1 cm.

3.A copper wire (α = 1,7.10-5K-1) whose length at 180C was 150 cm was heated by electric current and elongated by 9 mm. To what temperature was the wire heated?

Solution:

Analysis:

l0 = 150 cm = 1,5 m,   t1 = 180C,  Δl = 0,009 m,   α = 1,7.10-5K-1

 fyzika-tepelna-roztaznost-3.gif

The wire was heated to a temperature of 3710C.

4.Two rods, one iron (α1 = 1,2.10-5K-1) and one zinc (α2 = 2,9.10-5K-1), had the same length at 00C. When their temperature is raised to 1000C, the difference in their lengths is 1 cm. What were their original lengths?

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5.An aluminum container (α (Al) = 24.10-6K-1) has an internal volume of 10 liters at 200C. How does its internal volume change at 1000C?

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6.Tables state that copper has a density of ρ20 = 8930 kg.m-3 at 200C. What is the density of copper at 800C?

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7.A floor tile is made of marble (α = 0,85.10-5K-1) and is square. The side of the tile has a length of 0,5 m at 00C. By how many cm2 does the area of the tile increase if its temperature is raised to 350C?

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8.A brass sphere (α = 1,8.10-1K-1) has a radius r1 = 2 cm at 150C. By how many 0C must it be heated so that it will no longer pass through a circular opening with radius r2 = 2,02 cm?

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9.A tank wagon is filled with diesel up to the opening. (ρ = 940 kg.m-3, β = 1.10-3K-1) At 00C the wagon holds 50 tons of diesel. How much diesel will spill out through the opening if the temperature of the diesel rises to 200C during the trip?

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10. The air in a vessel with a movable piston has a volume of 1 liter at 100C. What is the volume of the air at 400C? Also calculate the relative change in the volume of the air!

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11.A student named “Mistaken” claimed that from the relation l = l0(1 + αΔt) it follows that with sufficiently large cooling the length “l” of a rod will be zero. Could he be right? Which important law of nature does this claim contradict?

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12. A 200 m long heating pipeline was welded at a temperature of 200C. By how much will it lengthen if, under operating conditions, the temperature increases to 1200C?

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13.  At a temperature of 100C a copper cube has a volume of 600 cm3. How will its volume change when heated to 2100C?

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14.Calculate the coefficient of volumetric thermal expansion of gasoline, which has a volume of 10.3·10-3 m3 at a temperature of 300C and a volume of 10-2 m3 at a temperature of 00C.

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15. Calculate the mass of 1 dm3 of copper at a temperature of 4000C, given that at 00C the density of copper is 8.9·103 kg·m-3.

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16.At 00C a zinc rod has a length of 200 mm and a copper rod has a length of 201 mm. Their cross-sectional dimensions at 00C are the same. Calculate at what temperature the rods will have the same volume.

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17.A steel sphere has a radius of 2 cm at a temperature of 800C. What is the volume of this sphere at a temperature of –200C?

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18.How much heat is absorbed by a copper rod that has a length of 10 cm and a cross-section of 2 cm2 at 200C, if it lengthens by 0.1 mm when heated?

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19.A small ball oscillates on a thin steel wire with a period of 2 seconds. The length of the suspension at 200C is 2 meters. How does the period change if the pendulum is heated to 800C?

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20.By how much will an aluminum wire with a cross-section of 5 mm2 lengthen if an electric current with power 16 W passes through it for 1 minute? (Neglect heat losses)

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