The subatomic structure of solids and liquids have a profound influence upon how they react to a transfer of thermal energy. Solids have a relatively fixed or rigid structure, whereas the molecular structure of liquids allows for greater expansion and compression. Measurements have shown that ( in general ) a linear relationship exists between the change in temperature ( T ) of a solid or liquid and its corresponding change in length in meters ( m ) or volume in cubic meters ( m^3 ).
Recall that the slope of a line has the general formula of F ( x ) = ( y ) = ( m )( x ) + b, where ( m ) is the slope of the line ( rise / run ). When temperature vs. expansion graphs are plotted for different substances, the slope of each line represents a constant of proportionality that is used in linear and volumetric thermal equations. The relative location of each slope shows how each substance resists expanding in response to increases ( or decreases ) in temperature. Note that 1 K degree is effectively the same size as 1 C degree. For this reason, net C degree temperature changes directly correspond to changes in K degrees. The equations of interest are Lf = ( 𝝰 )( Lo )( 𝝙 T ) and Vf = ( 𝞫)( Vo )( 𝝙 T ).
Q: Gasoline is placed into a steel container that holds 56 L. The temperature of the gasoline is 10 C. Inside the hot trunk of a car, the gasoline’s temperature soon rises to 20 C. How much gasoline ( if any ) will overflow out of the container? The temperature coefficient of volumetric expansion for gasoline is 𝞫= ( 950 * 10^-6 K^-1 ), and the volumetric expansion for steel is 𝞫 = ( 36 * 10^-6 K^-1 ).
A: We must first convert 56 L into an equivalent volume in meters ( m ). Since 1 mL = 1cc ( cm^3 ), this will be our conversion fraction: ( 56 L )( 1000 mL / 1 L )( 1 cc / 1 mL ) = 56 * 10^3 cm^3. Converting cm^3 to m^3, we have ( 56 * 10^3 cm^3 )( 1 m / 100 cm )^3 = 0.056 m^3.
The change in volume of the gasoline is as follows: Vf = ( 𝞫)( Vo )( 𝝙 T ) = ( 950 * 10^-6 K^-1 )( 0.056 m^3 )( + 10 K ) = + 5.32 * 10^-4 m^3. The change in volume of the steel container is Vf = ( 𝞫)( Vo )( 𝝙 T ) = ( 36 * 10^-6 K^-1 )( 0.056 m^3 )( + 10 K ) = + 0.202 * 10^-4 m^3. Therefore, about 0.51 * 10^-3 m^3 ( 0.51 L ) of gasoline overflows.
The y = ( m )( x ) + b plot for temperature vs. expansion is practical, but fine measurements show that the percent change that various materials undergo often change within temperature extremes. Further deviations from this norm are observed when the relationship between thermal energy and gas characteristics are studied.
RENAISSANCE PHYSICS 