Calculate Extra Gasoline in kg with Density and Expansion Rates

[cutegirl1980]

The density of gasoline is ρ = 0.730×103 kg/m^3 at 0ºC. Its average coefficient of volume expansion is α(sub v) = 9.60×10^-4 (ºC)-1. If 1.00 gallons of gasoline occupies 0.00380 m^3, how many extra kilograms of gasoline would you get if you bought 10.0 gallons of gasoline at 0ºC rather than at 20.0ºC from a gasoline pump that is not temperature compensated?

 

[Doc Al]

The definition of thermal volume expansion coefficient:
$$\Delta V/V_0 = \alpha_v \Delta T$$.

 

[wais]

To calculate the extra gasoline in kilograms, we need to first determine the volume of 10.0 gallons of gasoline at 0ºC and at 20.0ºC.

At 0ºC, 1 gallon of gasoline occupies 0.00380 m^3, so 10.0 gallons would occupy 0.0380 m^3.

At 20.0ºC, we need to take into account the expansion rate. Using the coefficient of volume expansion, we can calculate the increase in volume from 0ºC to 20.0ºC.

ΔV = α(sub v) * V * ΔT = (9.60×10^-4 (ºC)-1) * (0.0380 m^3) * (20.0ºC - 0ºC) = 0.007296 m^3

Therefore, at 20.0ºC, 10.0 gallons of gasoline would occupy 0.0380 m^3 + 0.007296 m^3 = 0.045296 m^3.

To calculate the extra gasoline in kilograms, we need to find the difference in volume between the two temperatures and then convert that to mass using the density of gasoline.

ΔV = 0.045296 m^3 - 0.0380 m^3 = 0.007296 m^3

Mass of extra gasoline = ΔV * ρ = (0.007296 m^3) * (0.730×103 kg/m^3) = 5.33 kg

Therefore, if you were to buy 10.0 gallons of gasoline at 0ºC rather than at 20.0ºC from a gasoline pump that is not temperature compensated, you would get an extra 5.33 kg of gasoline. It is important to note that this calculation assumes that the gasoline pump is not temperature compensated, meaning it does not adjust for the temperature of the gasoline. If the pump is temperature compensated, the extra gasoline would be negligible.