Question

Brewed coffee is often too hot to drink right away. You can cool it with an ice cube, but this dilutes it. Or you can buy a device that will cool your coffee without dilution - a 300 g aluminum cylinder that you take from your freezer and place in a mug of hot coffee. If the cylinder is cooled to -20∘C, a typical freezer temperature, and then dropped into a large cup of coffee (essentially water, with a mass of 500 g) at 85∘C, what is the final temperature of the coffee?

Answer 1

Specific heat of aluminum = C₁ = 900 J/(kg.^oC)

Specific heat of water = Specific heat of coffee = C₂ = 4186 J/(kg.^oC)

Mass of the aluminum cylinder = m₁ = 300 g = 0.3 kg

Mass of the coffee = m₂ = 500 g = 0.5 kg

Initial temperature of the aluminum cylinder = T₁ = -20 ^oC

Initial temperature of the coffee = T₂ = 85 ^oC

Final equilibrium temperature = T₃

The heat gained by the aluminum cylinder is equal to the heat lost by the coffee.

m₁C₁(T₃ - T₁) = m₂C₂(T₂ - T₃)

(0.3)(900)(T₃ - (-20)) = (0.5)(4186)(85 - T₃)

270T₃ + 5400 = 177905 - 2093T₃

2363T₃ = 172505

T₃ = 73 ^oC

Final temperature of the coffee = 73 ^oC