Final answer:
The final temperature of the mixed water is determined by setting the heat lost by the warmer water equal to the heat gained by the cooler water using the specific heat formula q=mcΔT, and solving for the final temperature, resulting in (75.0 g × 77.0°C + 100.0 g × 21.0°C) / (75.0 g + 100.0 g).
Step-by-step explanation:
To calculate the final temperature of mixed water, we must apply the principle of conservation of energy, specifically the concept of thermal equilibrium. The heat lost by the warmer water (75.0 g at 77.0°C) must be equal to the heat gained by the cooler water (100.0 g at 21.0°C) when they reach thermal equilibrium.
To compute this, we can use the formula q = mcΔT, where q is the heat absorbed or released, m is the mass of the water, c is the specific heat capacity of water (approximately 4.18 J/g°C), and ΔT is the change in temperature. Since the specific heat of water is constant, we can set the heat lost by the hot water equal to the heat gained by the cold water and solve for the final temperature.
The equation for the heat lost by the warm water is: q₁ = m₁c(Τ₂ - T₀). The equation for the heat gained by the cooler water is: q₂ = m₂c(T₀ - Τ₁). We set q₁ to be equal to q₂ and solve for the final temperature, T₀. After canceling out common factors and rearranging, the equation becomes: T₀ = (m₁Τ₂ + m₂Τ₁) / (m₁ + m₂). Plugging in the values, we get T₀ = (75.0 g × 77.0°C + 100.0 g × 21.0°C) / (75.0 g + 100.0 g), which results in a final temperature that can then be calculated.