Question

An aluminum piece has a volume of 20.0000 cm³ (this is a mass of 54.20 g ) at 200∘C. The hot aluminum piece is placed inside a graduate cylinder with 30.0000 cm³ (this is a mass of 22.287 g ) gasoline at 20∘C. a. What is the final temperature assuming no heat is transfer to the cylinder or the outside. The specific heat of aluminum is 0.900 J/(g∘C) and the specific heat of gasoline is 2.20 J/(g ∘C) b. Once the aluminum and gasoline are mixed and thermal equilibrium, what is the total volume reading on the graduated cylinder. Again, assume no heat is transferred to the cylinder or the outside. The linear expansion coefficient for aluminum is 24×10−6C⁻¹ and the volume expansion coefficient for gasoline is 9.6×10−4C⁻¹ .

Answer 1

a. The final temperature is found by equating the heat gained by aluminum to the heat lost by gasoline, using the formula m⋅Δm⋅c⋅ΔT.

b. The total volume in the graduated cylinder after reaching thermal equilibrium is the sum of the initial volumes and the changes in volumes due to thermal expansion of aluminum and gasoline.

Step-by-step explanation:

This problem involves the principles of heat transfer, thermal equilibrium, and volume expansion. Let's break it down into two parts:

Part a: Finding the final temperature

The heat gained by aluminum equals the heat lost by gasoline. The heat gained or lost (Q ) can be calculated using the formula:

Q=mcΔT

where:

-m is the mass,

- c is the specific heat,

- ΔT is the change in temperature.

For aluminum:

Q Aluminum=mAluminum ⋅cAluminum ⋅ΔTAluminum

For gasoline:

QGasoline =mGasoline ⋅c Gasoline ⋅ΔTGasoline

Since no heat is transferred to the cylinder or the outside, the total heat gained by aluminum is equal to the total heat lost by gasoline:

mAluminum ⋅c Aluminum ⋅ΔT Aluminum =m Gasoline ⋅c Gasoline​ ⋅ΔTGasoline

Now, rearrange the equation and solve for the final temperature.

Part b: Finding the total volume

Once the aluminum and gasoline reach thermal equilibrium, they will have the same final temperature. Use the volume expansion coefficients to find the change in volume for each substance:

ΔVAluminum=VAluminum⋅βAluminum⋅ΔTfinal

ΔVGasoline =VGasoline ⋅β Gasoline ⋅ΔT final

​

The total volume in the graduated cylinder after reaching thermal equilibrium is the sum of the initial volumes and the changes in volumes:

Vtotal=VAluminum, initial +VGasoline, initial +ΔVAluminum +ΔV Gasoline

​

Now, substitute the given values and solve for the total volume.