Question

You are given a pot of hot tea (350 mL) at a temperature of 85°C and a quantity of ice at -12°C. Determine the absolute minimum amount of ice you can add to the hot tea so that at equilibrium you have iced tea (ie. a very, very small amount of ice and some water). You can assume tea has the same density (1.00 g/mL) and specific heat (4190 J/kgK) as liquid water. The heat of fusion of H2O is 3.33x10^5 J/kg. The specific heat of ice is 2090 J/kgK.

Answer 1

348 g

Step-by-step explanation:

The heat gained by the ice equals the heat lost by the hot tea.

Also, since we require a very, very small amount of ice and some water, to still have some ice, the temperature of the hot tea has to decrease to the melting point of ice which is 0°C. So, the final temperature of the mixture is 0 °C.

So, mc(T - T') + mL = -MC(T - T") where m = mass of ice, c = specific heat of ice = 2090 J/kgK., T = final temperature of mixture = 0 °C, T' = initial temperature of ice = -12 °C, L = heat of fusion of H2O = 3.33 × 10⁵ J/kg, M = mass of hot tea = ρV where ρ = density of tea = 1.00 g/mL and V = volume of hot tea = 350 mL. So, M = ρV = 1.00 g/mL × 350 mL = 350 g = 0.350 kg, C = specific heat of tea = 4190 J/kgK and T" = initial temperature of tea = 85 °C.

Making m subject of the formula, we have

mc(T - T') + mL = -MC(T - T")

m[c(T - T') + L] = -MC(T - T")

m = -MC(T - T")/[c(T - T') + L]

substituting the values of the variables into the equation, we have

m = -MC(T - T")/[c(T - T') + L]

m = -0.350 kg × 4190 J/kgK(0 °C - 85 °C.)/[2090 J/kgK(0 °C. - (-12 °C)) + 3.33 × 10⁵ J/kg]

m = -1466.5 J/kK(- 85 °C)/[2090 J/kgK(0 °C + 12 °C)) + 3.33 × 10⁵ J/kg]

m = 124652.5 J/[2090 J/kgK(12 °C) + 3.33 × 10⁵ J/kg]

m = 124652.5 J/[25080 J/kg + 3.33 × 10⁵ J/kg]

m = 124652.5 J/358080 J/kg

m = 0.3481 kg

m = 348.1 g

m ≅ 348 g

So, the minimum amount of ice to be added is 348 g.