A 19.5-g sample of a metal was heated to 61.67°C. When the metal was placed into 26.5 g of water in a calorimeter, the temperature of the water increased from 25.00°C to 30.00°C. What is the specific heat of the metal?
[mass metal x specific heat metal x (Tfinal-Tinitial)] + [mass water x specific heat water x (Tfinal-Tinitial)] = 0
specific heat metal is the only unknown. Solve for that.
To calculate the specific heat of the metal, we can use the formula:
q = mcΔT
Where:
q = heat transfer
m = mass of the substance (water in this case)
c = specific heat capacity of the substance (unknown for the metal)
ΔT = change in temperature
First, let's calculate the heat transfer (q) for the water using the formula:
q = mcΔT
Given:
m (mass of water) = 26.5 g
c (specific heat of water) = 4.18 J/g°C (specific heat of water)
ΔT (change in temperature) = 30.00°C - 25.00°C = 5.00°C
Substituting these values into the formula:
q = (26.5 g)(4.18 J/g°C)(5.00°C)
q = 550.15 J
Now, let's calculate the heat transfer (q) for the metal. Since the metal was heated before being placed in the water, its temperature change is:
ΔT (change in temperature) = 61.67°C - 30.00°C = 31.67°C
Using the same formula:
q = mcΔT
Given:
m (mass of metal) = 19.5 g
c (specific heat of the metal) = unknown
ΔT (change in temperature) = 31.67°C
Substituting the known values:
550.15 J = (19.5 g)(c)(31.67°C)
Now, isolate c by dividing both sides of the equation by (19.5 g)(31.67°C):
c = 550.15 J / ((19.5 g)(31.67°C))
Calculating the value:
c ≈ 0.912 J/g°C
Therefore, the specific heat of the metal is approximately 0.912 J/g°C.
To find the specific heat of the metal, we need to apply the principle of energy conservation.
The heat gained by the metal is equal to the heat lost by the water. The heat gained by the metal can be calculated using the formula:
q = mcΔT
where:
q is the heat gained by the metal
m is the mass of the metal
c is the specific heat of the metal
ΔT is the change in temperature of the metal
The heat lost by the water can also be calculated using the same formula:
q = mcΔT
where:
q is the heat lost by the water
m is the mass of the water
c is the specific heat of water (which is 4.18 J/g°C, assuming water has a density of 1 g/mL)
ΔT is the change in temperature of the water
Since the heat gained by the metal is equal to the heat lost by the water, we can set these two equations equal to each other:
mcΔT (metal) = mcΔT (water)
We know the values for the mass of the metal, mass of the water, and the change in temperature of the water. We need to solve for the specific heat of the metal (c):
Let's substitute the known values into the equation:
19.5g * c * (61.67°C - 25.00°C) = 26.5g * 4.18 J/g°C * (30.00°C - 25.00°C)
Now we can solve for c:
19.5g * c * 36.67°C = 26.5g * 4.18 J/g°C * 5.00°C
Dividing both sides by (19.5g * 36.67°C):
c = (26.5g * 4.18 J/g°C * 5.00°C) / (19.5g * 36.67°C)
c = 0.361 J/g°C
Therefore, the specific heat of the metal is 0.361 J/g°C.