# Volume expansion – problems and solutions

1. At 30 ^{o}C the volume of an aluminum sphere is 30 cm^{3}. The coefficient of linear expansion is 24 x 10^{-6 o}C^{-1}. If the final volume is 30.5 cm^{3}, what is the final temperature of the aluminum sphere?

__Known :__

The coefficient of linear expansion (α) = 24 x 10^{-6 o}C^{-1}

The coefficient of volume expansion

^{-6 o}C

^{-1 }= 72 x 10

^{-6 o}C

^{-1}

The initial temperature (T_{1}) = 30^{o}C

The initial volume (V_{1}) = 30 cm^{3}

The final volume (V_{2}) = 30.5 cm^{3}

The change in volume (ΔV) = 30.5 cm^{3} – 30 cm^{3 }= 0.5 cm^{3}

__Wanted :__ The final temperature (T_{2})

__Solution :__

ΔV = β (V_{1})(ΔT)

ΔV = β (V_{1})(T_{2} – T_{1})

0.5 cm^{3 }= (72 x 10^{-6 o}C^{-1})(30 cm^{3})(T_{2} – 30^{o}C)

0.5 = (2160 x 10^{-6})(T_{2} – 30)

0.5 = (2.160 x 10^{-3})(T2 – 30)

0.5 = (2.160 x 10^{-3})(T_{2} – 30)

0.5 / (2.160 x 10^{-3}) = T_{2} – 30

0.23 x 10^{3} = T_{2} – 30

0.23 x 1000 = T_{2} – 30

230 = T_{2} – 30

230 + 30 = T_{2}

T_{2} = 260^{o}C

2. The coefficient of linear expansion of an metal sphere is 9 x 10^{-6 o}C^{-1}. The internal diameter of the metal sphere at 20 ^{o}C is 2.2 cm. If the final diameter is 2.8 cm, what is the final temperature!

__Known :__

The coefficient of linear expansion (α) = 9 x 10^{-6 o}C^{-1}

The coefficient of volume expansion (β) = 3 α = 3 x 9 x 10^{-6 o}C^{-1 }= 27 x 10^{-6 o}C^{-1}

The initial temperature (T_{1}) = 20^{o}C

The initial diameter

_{1}) = 2.2 cm

The final diameter (D_{2}) = 2.8 cm

The initial radius (r_{1}) = D_{1} / 2 = 2.2 cm^{3 }/ 2 = 1.1 cm^{3}

The final radius (r_{2}) = D_{2} / 2 = 2.8 cm^{3 }/ 2 = 1.4 cm^{3}

The initial volume (V_{1}) = 4/3 π r_{1}^{3} = (4/3)(3.14)(1.1 cm)^{3} = (4/3)(3.14)(1.331 cm^{3}) = 5.57 cm^{3}

The final volume (V_{2}) = 4/3 π r_{2}^{3} = (4/3)(3.14)(1.4 cm)^{3} = (4/3)(3.14)(2.744 cm^{3}) = 11.48 cm^{3}

The change in volume (ΔV) = 11.48 cm^{3 }– 5.57 cm^{3 }= 5.91 cm^{3 }

__Wanted :__ The final temperature (T_{2})

__Solution :__

ΔV = β (V_{1})(ΔT)

5.91 cm^{3 }= (27 x 10^{-6 o}C^{-1})(5.57 cm^{3})(T_{2} – 20^{o}C)

5.91 = (150.39 x 10^{-6})(T_{2} – 20)

5.91 / 150.39 x 10^{-6 }= T_{2} – 20

0.039 x 10^{6} = T_{2} – 20

39 x 10^{3} = T_{2} – 20

39,000 = T_{2} – 20

39,000 + 20 = T

_{2}

T_{2 }= 39,020 ^{o}C

3. A 2000-cm^{3 }aluminum container, filled with water at 0^{o}C. And then heated to 90^{o}C. If the coefficient of linear expansion for aluminum is 24 x 10^{-6 }(^{o}C)^{-1 }and the coefficient of volume expansion for water is 6.3 x 10^{-4} (^{o}C)^{-1}, determine the volume of spilled water.

__Known :__

The initial volume of the aluminum container and water (V_{o}) = 2000 cm^{3} = 2 x 10^{3 }cm^{3}

The initial temperature of the aluminum container and water (T_{1}) = 0^{o}C

The final temperature of the aluminum container and water (T_{2}) = 90^{o}C

The coefficient of linear expansion for aluminum (α) = 24 x 10^{-6 }(^{o}C)^{-1 }

The coefficient of volume expansion for aluminum (γ) = 3α = 3 (24 x 10^{-6 }(^{o}C)^{-1 }) = 72 x 10^{-6} ^{o}C^{-1}

The coefficient of volume expansion for water (γ) = 6.3 x 10^{-4} (^{o}C)^{-1}

__Wanted :__ The volume of spilled water

__Solution :__

The equation of the volume expansion :

V = V_{o} + γ V_{o} ΔT

V – V_{o} = γ V_{o} ΔT

ΔV = γ V_{o} ΔT

*V = final volume, **V*_{o}* = initial volume, **ΔV = the change in volume, **γ = the coefficient of volume expansion, **ΔT = the change in temperature*

Calculate the change in volume of the aluminum container :

ΔV = γ V_{o} ΔT = (72 x 10^{-6})(2 x 10^{3})(90) = 12960 x 10^{-3} = 12.960 cm^{3 }

Calculate the change in volume of the water :

ΔV = γ V_{o} ΔT = (6.3 x 10^{-4})(2 x 10^{3})(90) = 1134 x 10^{-1} = 113.4 cm^{3}

*The change in volume of the water is more greater than the aluminum container so that some water spilled.*

Calculate the volume of spilled water :

113.4 cm^{3} – 12.960 cm^{3 }= 100.44 cm^{3}

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- Converting temperature scales
- Linear expansion
- Area expansion
- Volume expansion
- Heat
- Mechanical equivalent of heat
- Specific heat and heat capacity
- Latent heat, the heat of fusion, the heat of vaporization
- Energy conservation for heat transfer