After 6.00 kg of water at 77.3 oC is mixed in a perfect thermos with 3.00 kg of ice at 0.0 oC, the mixture is allowed to reach equilibrium. When heat is added to or removed from a solid or liquid of mass m and specific heat capacity c, the change in entropy can be shown to be ΔS = mc ln(Tf/Ti), where Ti and Tf are the initial and final Kelvin temperatures. Using this expression and the change in entropy for melting, find the change in entropy that occurs.
First you will have to compute the final temperature and determine if all the ice melts. (It does, in this case)
Use the formula they gave you for the entropy lost by the cooling liquid water.
The entropy gained by the melting ice is [80 cal/g]*3000 g/273K
There will be additional entropy gain
ÄS = m c ln(Tf/Ti),
as the melted ice incgeases to the final temperature Tf.
Ti = 273 K m(ice) = 3000 g
Add up all the entropy gains and losses for the final answer.
To find the change in entropy that occurs when the given quantities of water and ice reach equilibrium, we need to consider the change in entropy for both the heat transfer from the water and the phase change of the ice.
1. First, let's consider the change in entropy due to the heat transfer from the water. We are given the mass of water (m = 6.00 kg), the specific heat capacity of water (c = 4.18 J/g·°C), and the initial and final temperatures in Celsius (Ti = 77.3°C and Tf).
Step 1: Convert the temperatures to Kelvin
Ti = 77.3°C + 273.15 = 350.45 K
Tf = Convert "Tf" in Celsius to Kelvin
Step 2: Calculate the change in entropy using the given formula.
ΔS_water = mc ln(Tf/Ti)
Convert the mass from kg to grams:
m = 6.00 kg × 1000 = 6000 g
Substitute the values:
ΔS_water = (6000 g) × (4.18 J/g·°C) × ln(Tf/Ti)
Now we will move on to the change in entropy due to the phase change of the ice.
2. The phase change of the ice from solid to liquid occurs at 0°C, and we know the mass of the ice (m = 3.00 kg) and the change in entropy for melting, which is given.
Step 1: Convert the given change in entropy for melting to Kelvin.
The change in entropy for melting ice is given by:
ΔS_melt = ΔH_melt / T_melt
The heat of fusion for water, ΔH_melt, is 333.55 J/g. To convert this to Kelvin, we need to divide by the molar mass of water (18.015 g/mol):
ΔS_melt = (333.55 J/g) / (18.015 g/mol) = 18.51 J/mol·K
Step 2: Calculate the change in entropy due to the phase change of the ice.
ΔS_ice = ΔS_melt × (mass ice / molar mass ice)
= (18.51 J/mol·K) × (3000 g / 18.015 g/mol)
3. Finally, calculate the total change in entropy by adding the change in entropy due to heat transfer from the water and the change in entropy due to the phase change of the ice.
ΔS_total = ΔS_water + ΔS_ice
Substitute the calculated values to find the final answer.