A 100-g chunk of 72 ∘C iron is dropped into a cavity in a very large block of ice at 0∘C.
What is the mass of ice that melts. (The specific heat capacity of iron is 0.11 cal/g⋅∘C.).
heat of fusion for ice ... 79.7 cal/g
heat from iron
... 100 g * 72 ºC * 0.11 cal/g⋅ºC
divide heat from iron by heat of fusion
Well, that's a cool question! Let's break it down.
First of all, we need to calculate the heat gained by the ice to melt. We can use the formula:
Q = m * c * ΔT,
where Q is the heat gained, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
Since the ice is at 0∘C and the iron is at 72∘C, the change in temperature is 72∘C - 0∘C = 72∘C.
Now, we know the specific heat capacity of iron is 0.11 cal/g⋅∘C, and the mass of the iron is 100 grams.
Let's plug in the numbers:
Q = 100 g * 0.11 cal/g⋅∘C * 72∘C.
Calculating that would give us the heat gained by the ice. But hold on a second, we actually want to know the mass of ice that melts. Silly me!
The heat gained by the ice would be equal to the heat lost by the iron. So we can use the same formula to calculate the mass of the ice:
Q = m * c * ΔT.
We'll make m the subject of the formula:
m = Q / (c * ΔT).
Now we can plug in the numbers and calculate the mass of the ice that melts. Ahh, the suspense is melting away!
To calculate the mass of ice that melts, we need to use the principle of heat transfer. The heat gained by the ice must be equal to the heat lost by the iron.
The heat gained by the ice is given by the equation: Q = m × c × ΔT
Where:
Q = Heat gained or lost
m = Mass
c = Specific heat capacity
ΔT = Change in temperature
In this case, the ice gains heat, so the equation becomes:
Q_ice = m_ice × c_ice × ΔT_ice
The heat lost by the iron is given by the equation: Q = m × c × ΔT
Where:
Q = Heat gained or lost
m = Mass
c = Specific heat capacity
ΔT = Change in temperature
In this case, the iron loses heat, so the equation becomes:
Q_iron = m_iron × c_iron × ΔT_iron
Since the heat gained by the ice is equal to the heat lost by the iron, we can set up an equation:
Q_ice = Q_iron
m_ice × c_ice × ΔT_ice = m_iron × c_iron × ΔT_iron
Given:
m_iron = 100 g
c_iron = 0.11 cal/g⋅∘C
ΔT_iron = (0∘C) - (72∘C) = -72∘C
We also know that the final temperature of both the iron and the ice will be 0∘C.
So, ΔT_ice = (0∘C) - (0∘C) = 0∘C
Plugging in the values:
m_ice × c_ice × 0∘C = 100 g × 0.11 cal/g⋅∘C × (-72∘C)
Simplifying,
0 = 100 g × 0.11 cal/g⋅∘C × (-72∘C)
0 = -792 g⋅cal
Since the left side of the equation is zero, we can conclude that the mass of ice that melts is also zero.
Therefore, no ice will melt when a 100-g chunk of 72∘C iron is dropped into a cavity in a very large block of ice at 0∘C.
To find the mass of ice that melts, we need to calculate the heat transferred from the iron to the ice using the specific heat capacities of the substances involved.
The heat transferred from the iron to the ice can be calculated using the formula:
Q = mcΔT
Where:
Q is the heat transferred
m is the mass of the substance
c is the specific heat capacity of the substance
ΔT is the change in temperature
First, let's find the heat transferred from the iron:
Q_iron = mcΔT
Q_iron = (100 g)(0.11 cal/g⋅∘C)(72 ∘C - 0 ∘C)
Q_iron = 792 cal
Since energy is conserved, the heat transferred from the iron will be equal to the heat absorbed by the ice. Assuming there is no heat loss to the surroundings, the heat absorbed by the ice is equal to the heat released by the iron:
Q_ice = Q_iron
Q_ice = 792 cal
Now, let's find the mass of ice that melts. We know that the heat required to melt 1 gram of ice is 1 calorie. Therefore, the mass of ice that melts can be calculated by dividing the heat absorbed by the ice by the heat required to melt 1 gram of ice:
Mass of ice that melts = Q_ice / Heat required to melt 1 gram of ice
Mass of ice that melts = 792 cal / 1 cal/g
Mass of ice that melts = 792 g
Therefore, the mass of ice that melts is 792 grams.