A 240 g ice cube at -10 C is placed in an aluminum cup whose initial temperature is 70 C. The system comes to an equilibrium temperature of 20 C.
What is the mass of the cup?(answer in kg)
Write an equation that says the heat gained by the ice equals the heat lost by the cup. The only unknown will be the aluminum mass, which you can solve for. You will need to look up the specific heats of aluminum and ice. The heat of fusion of ice is 80 calories per gram, and will be part of the equation.
Show your work if you need further assistance.
To find the mass of the cup, we can use the principle of conservation of energy, specifically the equation:
Qice + Qcup = 0
Where Qice is the heat gained by the ice cube and Qcup is the heat gained by the aluminum cup. Since the equilibrium temperature is lower than the initial temperature of the cup, we know that the cup gains heat from the ice cube.
First, let's find the heat gained by the ice cube. We use the equation:
Qice = m × c × ΔT
Where
m is the mass of the ice cube (240 g = 0.24 kg),
c is the specific heat capacity of ice (-2.1 kJ/kg°C, or -2100 J/kg°C),
ΔT is the change in temperature (-10°C - 20°C = -30°C).
Plugging in the values, we get:
Qice = 0.24 kg × -2100 J/kg°C × -30°C = 15,120 J
Since Qice and Qcup are equal, we can now find the heat gained by the cup. We use the equation:
Qcup = m × c × ΔT
Where
m is the unknown mass of the cup in kg,
c is the specific heat capacity of aluminum (0.9 kJ/kg°C, or 900 J/kg°C),
ΔT is the change in temperature (70°C - 20°C = 50°C).
Plugging in the values, we get:
15,120 J = m × 900 J/kg°C × 50°C
Simplifying the equation gives:
15,120 J = 45,000 J/kg × m
To solve for m, divide both sides of the equation by 45,000 J/kg:
m = 15,120 J / 45,000 J/kg ≈ 0.336 kg
Therefore, the mass of the cup is approximately 0.336 kg.