What will be the equilibrium temperature when a 273 g block of copper at 298°C is placed in a 132 g aluminum calorimeter cup containing 815 g of water at 11.4°C?
To find the equilibrium temperature, we can use the principle of heat transfer, which states that the heat lost by one object is equal to the heat gained by another object in thermal contact with it.
First, let's calculate the heat lost by the copper block (q1). We have the mass of the copper block (273 g) and its initial temperature (298 °C). The specific heat capacity of copper is 0.39 J/g°C. We can use the formula:
q1 = m1 * c1 * ΔT1
where q1 is the heat lost, m1 is the mass of the copper block, c1 is the specific heat capacity of copper, and ΔT1 is the change in temperature (final temperature - initial temperature).
q1 = 273 g * 0.39 J/g°C * (T - 298 °C)
Next, let's calculate the heat gained by the water and the aluminum cup combined (q2). The mass of the water is 815 g, while the mass of the aluminum cup is 132 g. The specific heat capacity of water is 4.18 J/g°C, and the specific heat capacity of aluminum is 0.9 J/g°C. We can use the following formula:
q2 = (m2 * c2 * ΔT2) + (m3 * c3 * ΔT3)
where q2 is the heat gained, m2 is the mass of the water, c2 is the specific heat capacity of water, ΔT2 is the change in temperature of the water, m3 is the mass of the aluminum cup, c3 is the specific heat capacity of aluminum, and ΔT3 is the change in temperature of the cup.
q2 = (815 g * 4.18 J/g°C * (T - 11.4 °C) + (132 g * 0.9 J/g°C * (T - 11.4 °C)
Finally, since the heat lost by the copper block is equal to the heat gained by the water and the aluminum cup, we can set up an equation:
q1 = q2
273 g * 0.39 J/g°C * (T - 298 °C) = (815 g * 4.18 J/g°C * (T - 11.4 °C) + (132 g * 0.9 J/g°C * (T - 11.4 °C)
Now, we can solve this equation to find the equilibrium temperature (T). By rearranging the equation, we get:
273 g * 0.39 J/g°C * T - 273 g * 0.39 J/g°C * 298 °C = (815 g * 4.18 J/g°C * T - 815 g * 4.18 J/g°C * 11.4 °C + 132 g * 0.9 J/g°C * T - 132 g * 0.9 J/g°C * 11.4°C
Simplifying the equation further:
0.39 J/g°C * T - 0.39 J/g°C * 298 °C = 4.18 J/g°C * T - 4.18 J/g°C * 11.4 °C + 0.9 J/g°C * T - 0.9 J/g°C * 11.4°C
Now, we can combine like terms:
0.39 T - 116.22 = 4.18 T - 47.652 + 0.9 T - 10.26
We can further simplify the equation:
5.47 T - 116.22 = 4.18 T - 57.912
Now, we can solve for T:
5.47 T - 4.18 T - 116.22 + 57.912 = 0
1.29 T - 58.308 = 0
1.29 T = 58.308
T = 58.308 / 1.29
T ≈ 45.16 °C
Therefore, the equilibrium temperature will be approximately 45.16 °C.