50 grams of water at 85ºC were passed through one side of a two-compartment heat
exchanger. Sixty-five grams of 30ºC were passed through the other side. Both
streams of water came out at 50ºC Calculate the amount of heat given up by the hot water and the amount absorbed by the cold water. Why are there not the same?
To calculate the amount of heat given up by the hot water and absorbed by the cold water, we can use the formula:
Q = m * c * ΔT
Where:
Q is the amount of heat
m is the mass of the water
c is the specific heat capacity of water
ΔT is the change in temperature
Let's first calculate the heat given up by the hot water:
Mass of hot water (m1) = 50 grams
Initial temperature of hot water (T1) = 85ºC
Final temperature of hot water (T2) = 50ºC
Specific heat capacity of water (c) = 4.18 J/gºC
Change in temperature (ΔT) = T2 - T1 = 50ºC - 85ºC = -35ºC (negative because temperature is decreasing)
Using the formula:
Q1 = m1 * c * ΔT
= 50 g * 4.18 J/gºC * -35ºC
= -7305 J (negative sign indicates heat loss)
So, the amount of heat given up by the hot water is -7305 J (approximately).
Now let's calculate the heat absorbed by the cold water:
Mass of cold water (m2) = 65 grams
Initial temperature of cold water (T3) = 30ºC
Final temperature of cold water (T4) = 50ºC
Change in temperature (ΔT) = T4 - T3 = 50ºC - 30ºC = 20ºC
Using the formula:
Q2 = m2 * c * ΔT
= 65 g * 4.18 J/gºC * 20ºC
= 5422 J
So, the amount of heat absorbed by the cold water is 5422 J (approximately).
Now, why are the amounts of heat given up and absorbed not the same?
In a heat exchanger, the hot and cold fluids flow on opposite sides of the exchanger and exchange heat through a conductive barrier. The heat transfer occurs due to the temperature difference between the hot and cold fluids. The heat flows from the hot fluid to the cold fluid until they reach a common temperature.
In this case, the hot water initially had a higher temperature (85ºC) and the cold water had a lower temperature (30ºC). As they flowed through the heat exchanger, heat was transferred from the hot water to the cold water until they both reached the same final temperature (50ºC).
The difference in the initial temperatures of the hot and cold water resulted in a temperature gradient that facilitated the heat transfer. As a result, the hot water gave up a certain amount of heat energy, and the cold water absorbed the same amount of heat energy. However, due to the different masses of the hot and cold water and their specific heat capacities, the actual quantity of heat transferred may differ. This is why the amounts of heat given up and absorbed are not the same.