A 37.0 g sample of copper at 99.8 °C is carefully placed into an insulated container containing 182 g of water at 18.5 °C. Calculate the final temperature when thermal equilibrium is reached.

Assume there is no energy transferred to or from the container.

Specific heat capacities:
- Cu = 0.385 J g⁻¹ °C⁻¹
- H₂O = 4.184 J g⁻¹ °C⁻¹

To calculate the final temperature when thermal equilibrium is reached, we can use the principle of energy conservation. The heat lost by the copper sample is equal to the heat gained by the water. The equation we can use is:

Q_lost = Q_gained

The heat lost by the copper sample (Q_lost) can be calculated using the equation:

Q_lost = m_cu * c_cu * ΔT_cu

where m_cu is the mass of the copper, c_cu is the specific heat capacity of copper, and ΔT_cu is the change in temperature of the copper.

The heat gained by the water (Q_gained) can be calculated using the equation:

Q_gained = m_water * c_water * ΔT_water

where m_water is the mass of the water, c_water is the specific heat capacity of water, and ΔT_water is the change in temperature of the water.

Since there is no energy transferred to or from the container, we can assume that the heat lost by the copper is equal to the heat gained by the water. Therefore, we have:

m_cu * c_cu * ΔT_cu = m_water * c_water * ΔT_water

Plugging in the given values:

(37.0 g) * (0.385 J/g°C) * (T_f - 99.8°C) = (182 g) * (4.184 J/g°C) * (T_f - 18.5°C)

Simplifying the equation:

14.245 (T_f - 99.8) = 762.688 (T_f - 18.5)

Solving for T_f:

14.245T_f - 14.245(99.8) = 762.688T_f - 762.688(18.5)

14.245T_f - 1420.411 = 762.688T_f - 14137.608

14.245T_f - 762.688T_f = -14137.608 + 1420.411

-748.443T_f = -12717.197

T_f = -12717.197 / -748.443

T_f ≈ 16.98°C

Therefore, the final temperature when thermal equilibrium is reached is approximately 16.98°C.

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