Question

Linear Thermal Expansion (in one dimension)

1) The change in length ΔL is proportional to the original length L, and the change in temperature ΔT : ΔL = αLΔT, where ΔL is the change in length , and α is the coefficient of linear expansion.
a) The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest (–15ºC). The coefficient of linear expansion, α , for steel is 12×10−6 /ºC. When the temperatures rises to 25 °C, what is its change in length in meters?
2) The change in volume ΔV is very nearly ΔV ≈ 3αVΔT . This equation is usually written as ΔV = βVΔT, where β is the coefficient of volume expansion and β ≈ 3α . V is the original volume. ΔT is the change in temperature. Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gasoline, and both the tank and the gasoline have a temperature of 15.0ºC . The coefficients of volume expansion, for gasoline is βgas = 950×10−6 /ºC , for the steel tank is βsteel = 35×10−6 /ºC .
a) What is the change in volume (in liters) of the gasoline when the temperature rises to 25 °C in L?
b) What is the change in volume (in liters) of the tank when the temperature rises to 25 °C in L?
c) How much gasoline would be spilled in L?

Answer 1

1)
`\Delta L= 0.612\ m`

2) a.
`\Delta V_G=0.57\ L`

b.
`\Delta V_S=0.021\ L`

c.
`V_0=0.549\ L`

Step-by-step explanation:

1)

• given initial length,
  `L=1275\ m`

• initial temperature,
  `T_i=-15^(\circ)C`

• final temperature,
  `T_f=25^(\circ)C`

• coefficient of linear expansion,
  `\alpha=12* 10^(-6)\ ^(\circ)C^(-1)`

∴Change in temperature:

`\Delta T=T_f-T_i`

`\Delta T=25-(-15)`

• 
  `\Delta T=40^(\circ)C`

We have the equation for change in length as:

`\Delta L= L.\alpha. \Delta T`

`\Delta L= 1275* 12* 10^(-6)* 40`

`\Delta L= 0.612\ m`

2)

Given relation:

`\Delta V=V.\beta.\Delta T`

where:

`\Delta V`= change in volume

V= initial volume

`\Delta T`=change in temperature

• initial volume of tank,
  `V_(Si)=60\ L`

• initial volume of gasoline,
  `V_(Gi)=60\ L`

• initial temperature of steel tank,
  `T_(Si)=15^(\circ)C`

• initial temperature of gasoline,
  `T_(Gi)=15^(\circ)C`

• coefficients of volumetric expansion for gasoline,
  `\beta_G=950* 10^(-6)\ ^(\circ)C`

• coefficients of volumetric expansion for gasoline,
  `\beta_S=35* 10^(-6)\ ^(\circ)C`

a)

final temperature of gasoline,
`T_(Gf)=25^(\circ)C`

∴Change in temperature of gasoline,

`\Delta T_G=T_(Gf)-T_(Gi)`

`\Delta T_G=25-15`

`\Delta T_G=10^(\circ)C`

Now,

`\Delta V_G= V_G.\beta_G.\Delta T_G`

`\Delta V_G=60* 950* 10^(-6)* 10`

`\Delta V_G=0.57\ L`

b)

final temperature of tank,
`T_(Sf)=25^(\circ)C`

∴Change in temperature of tank,

`\Delta T_S=T_(Sf)-T_(Si)`

`\Delta T_S=25-15`

`\Delta T_S=10^(\circ)C`

Now,

`\Delta V_S= V_S.\beta_S.\Delta T_S`

`\Delta V_S=60* 35* 10^(-6)* 10`

`\Delta V_S=0.021\ L`

c)

Quantity of gasoline spilled after the given temperature change:

`V_0=\Delta V_G-\Delta V_S`

`V_0=0.57-0.021`

`V_0=0.549\ L`