A quantity of water at 87 degrees C is cooled at a rate of 2844 MJ/h. After 40 minutes, the water has changed to ice, having a temperature of 243K. Calculate the mass of the water/ice.
To calculate the mass of the water/ice, we will use the concept of specific heat capacity and the formula:
Q = mcΔT
Where:
Q = Heat transfer (energy)
m = Mass of the substance
c = Specific heat capacity
ΔT = Change in temperature
In this case, we have two stages: cooling the water and then freezing it to ice. Let's break down the calculation step by step:
1. Calculate the heat transfer during the cooling phase:
Given:
Initial temperature of water (Ti) = 87 degrees C
Final temperature of water (Tf) = 0 degrees C
Rate of cooling (Q) = 2844 MJ/h
Time (t) = 40 minutes
First, convert the temperature change from Celsius to Kelvin:
ΔT = Tf - Ti
ΔT = 0 - 87 = -87 degrees C
ΔT in Kelvin = -87 + 273.15 = 186.15 K
Next, convert the time from minutes to hours:
t = 40 minutes / 60 minutes/hour = 2/3 hour.
Now we can calculate the heat transfer during the cooling phase:
Q1 = Q * t
Q1 = 2844 MJ/h * (2/3) h = 1896 MJ
2. Calculate the mass of water using specific heat capacity:
The specific heat capacity of water is approximately 4.18 J/g*K.
Since energy is given in MJ (mega-joules), we need to convert it to joules:
1 MJ = 10^6 J
Q1 in joules = Q1 * 10^6 = 1896 MJ * 10^6 = 1896 * 10^6 J
Now we can rearrange the formula to solve for mass:
m = Q / (c * ΔT)
m = (Q1 in joules) / (c * ΔT)
m = (1896 * 10^6 J) / (4.18 J/g*K * 186.15 K)
3. Calculate the mass of ice:
Since the water has now changed to ice, we need to consider the heat transfer during freezing. The specific heat capacity of ice is approximately 2.09 J/g*K.
Given:
Final temperature of ice (Tfi) = 243 K
Change in temperature during freezing (ΔTf) = Tf - Tfi
ΔTf = 0 - 243 = -243 K
Now we can calculate the heat transfer during the freezing phase:
Q2 = mcΔT
Q2 = m * (2.09 J/g*K) * (-243 K)
Finally, calculate the mass of ice:
m_ice = Q2 / (-2.09 J/g*K)
Once you plug in the values and perform the calculations, you will obtain the mass of the water/ice.