Question

A \(10.0-g\) ice cube at \(-10.0^{\circ} \mathrm{C}\) is dropped into \(40.0 \mathrm{~g}\) of water at \(30.0^{\circ} \mathrm{C}\) a) After enough time has passed to allow the ice cube and water to come to equilibrium, what is the temperature of the water? b) If a second ice cube is added, what will the temperature be?

Short answer

Answer: The final equilibrium temperature after adding two ice cubes is 0°C.

Step-by-step solution

Calculate heat to raise the temperature of ice to 0°C

First, we need to calculate the amount of heat required to raise the temperature of the ice cube from -10°C to 0°C. We'll use the formula Q = mcΔT, where Q is the heat, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. For this step, we have: Mass of ice (m) = 10 g Initial temperature (T1) = -10°C Final temperature (T2) = 0°C Specific heat of ice (c) = 2.1 J/g°C Thus, ΔT = T2 - T1 = 0°C - (-10°C) = 10 °C Heat required (Q) = mcΔT = (10 g)(2.1 J/g°C)(10°) = 210 J

Calculate heat absorbed during melting of ice

Next, we need to calculate the heat absorbed by the ice during the melting process. We'll use the formula Q = mL, where Q is the heat, m is the mass, and L is the latent heat of fusion. For this step, we have: Latent heat of fusion (L) = 334 J/g Heat absorbed (Q) = mL = (10 g)(334 J/g) = 3340 J

Calculate heat lost by water to ice

The heat lost by the water to the ice is equal to the sum of the heat gained by the ice in Step 1 and Step 2. Therefore, Q_water = Q_ice1 + Q_ice2 = 210 J + 3340 J = 3550 J.

Calculate the final temperature after adding the first ice cube

The heat lost by water equals the product of its mass, specific heat capacity and the change in temperature. We'll use the formula Q = m_water c_water ΔT. For this step, we have: Mass of water (m_water) = 40 g Specific heat of water (c_water) = 4.18 J/g°C Initial temperature of water (T1_water) = 30°C 3550 J = (40 g)(4.18 J/g°C)(T2_water - 30°C) Now solve for T2_water: 3550 J / (40 g)(4.18 J/g°C) + 30°C ≈ 9.96°C Thus, the final temperature after adding the first ice cube is approximately 9.96°C.

Calculate the final temperature after adding the second ice cube

For this step, we can assume that the second ice cube has the same mass and initial temperature as the first one. Since the final temperature after adding the first ice cube is below the melting point of ice, the second ice cube will not melt completely. Therefore, the final temperature will remain at 0°C.