A vessel contains 10 kg of water and an unknown mass of ice at 0°C. The temperature of the vessel is monitored for 60 minutes. During the first 50 minutes, the temperature remains at 0°C. For the next 10 minutes, the temp steadily increases from 0°C to 2°C.
a. Find the rate that heat is entering the vessel.
To find the rate that heat is entering the vessel, we need to calculate the amount of heat transferred during the 10 minutes when the temperature increases from 0°C to 2°C.
The amount of heat transferred can be calculated using the formula:
Q = mcΔT
Where:
Q = amount of heat transferred
m = mass of the substance (in this case, water + ice)
c = specific heat capacity of the substance
ΔT = change in temperature
First, we need to determine the mass of the ice. The total mass of the system is given as 10 kg, and since all the water is at 0°C, it must be in the form of ice. Therefore, the mass of the ice is 10 kg.
Next, we need to find the specific heat capacity of ice. The specific heat capacity of water is typically given as 4.18 J/g°C or 4.18 kJ/kg°C. However, the specific heat capacity of ice is slightly different, so we need to look up this value. Let's assume the specific heat capacity of ice is 2.09 kJ/kg°C.
Now we can calculate the amount of heat transferred:
Q = mcΔT
= (10 kg) * (2.09 kJ/kg°C) * (2°C - 0°C)
= 41.8 kJ
The amount of heat transferred during the 10 minutes is 41.8 kJ.
To find the rate of heat transfer, we divide the amount of heat transferred by the time:
Rate of heat transfer = Q / t
= 41.8 kJ / 10 min
= 4.18 kJ/min
Therefore, the rate that heat is entering the vessel is 4.18 kJ/min.
To find the rate at which heat is entering the vessel, we need to calculate the heat gained by the ice during the 10 minutes when the temperature increases from 0°C to 2°C.
The heat gained or lost by an object can be calculated using the equation:
Q = m * c * ΔT
where Q is the heat gained or lost, m is the mass of the object, c is the specific heat capacity, and ΔT is the change in temperature.
In this case, the mass of the ice is unknown, but we can assume that it remains constant throughout the process since no water is added or removed. Let's represent the mass of the ice as "m" kg.
Given that the temperature of the ice increases from 0°C to 2°C, ΔT is equal to 2°C - 0°C = 2°C.
The specific heat capacity of ice (c) is 2.09 J/g°C, which can be converted to J/kg°C by multiplying by 1000 since there are 1000 grams in a kilogram.
c = 2.09 J/g°C * 1000 g/kg = 2090 J/kg°C
Now we can calculate the rate that heat is entering the vessel by dividing the heat gained during the 10 minutes by the time it took (10 minutes).
Rate of heat entering the vessel = Q / t
Q = m * c * ΔT
Q = m * 2090 J/kg°C * 2°C
Q = 4180 m J
Rate of heat entering the vessel = (4180 m J) / (10 minutes)
Rate of heat entering the vessel = 418 m J/min
Therefore, the rate that heat is entering the vessel is 418 J/min.