You want to quickly set up a temporary water bath in your lab with a volume of 10.0 L and a temperature of 37.0 °C. You only have hot water from your hot water faucet (temperature = 59.8 °C) and cold water from your cold water faucet (temperature = 22.0 °C). What volume of hot water (in liters) must you mix with cold water to get 10.0 L of 37.0 °C water? Assume the specific heat of the water is 4.184 J/g･K and that the water has a density of 1.00 g/mL.

The setup of the temporary water bath requires an understanding of the principle of heat exchange. This principle states that the heat gained by the cold water is equal to the heat lost by the hot water. 1) First, we'll start with the known information: - The final volume (Vf) is 10.0 L. - The final temperature (Tf) is 37.0 °C. - The initial temperature of the hot water (Th) is 59.8 °C. - The initial temperature of the cold water (Tc) is 22.0 °C. - The specific heat capacity (c) of water is 4.184 J/g･K. - The density (d) of water is 1.00 g/mL. 2) We want to find out the volume (Vh) of hot water we need to mix with cold water. Let's denote the volume of cold water as Vc. 3) According to the principle of heat exchange and the law of conservation of matter: - The heat lost by the hot water = Heat gained by the cold water. - The total volume of mixture is the sum of the volumes of the hot and cold water (Vf = Vh + Vc). 4) Let's express the heat exchange equation in terms of the known quantities and what we want to find: - Quantity of heat is given by mass (m) times specific heat capacity (c) times change in temperature (ΔT): q = mcΔT. - Because density (d) is mass per unit volume (m = dV), we can replace m with dV in the equation: q = dVcΔT. - Let's express the heat lost by the hot water and the heat gained by the cold water: d * Vh * c * (Th - Tf) = d * Vc * c * (Tf - Tc). - We can cancel out the densities and specific heat capacities on both sides. 5) From (Vf = Vh + Vc), we can also express Vc in terms of Vf and Vh: Vc = Vf - Vh. 6) Now we substitute Vc into the heat exchange equation with our known values: - Vh * (Th - Tf) = (Vf - Vh) * (Tf - Tc). 7) Let's plug in the known values: - Vh * (59.8°C - 37.0°C) = (10.0L - Vh) * (37.0°C - 22.0°C). - Vh * 22.8 = 10.0 * 15 - 15 * Vh. 8) Solving the above equation for Vh by combining the 'Vh' terms and then dividing by the coefficient of Vh gives: - Vh = (10.0 * 15) / (22.8 + 15) ≈ 6.15 L. So, to achieve a 10.0 L mixture at a temperature of 37.0 °C, you should mix approximately 6.15 L of hot water with the rest being cold water.