If 100g of zinc (c=377J/kg degrees celcius) at 95 degrees celcius is dropped into 60g of water contained in a 100g calorimeter (c=419J/kg degrees celcius) at 20 degrees celcius, what will the final temperature be?
To find the final temperature of the system, we need to calculate the heat lost by the zinc and the heat gained by the water and the calorimeter.
Step 1: Calculate the heat lost by the zinc:
Q1 = m1 * c1 * ΔT1
where,
m1 = mass of zinc = 100g
c1 = specific heat capacity of zinc = 377 J/kg·°C
ΔT1 = change in temperature = final temperature - initial temperature
The initial temperature of the zinc is 95°C.
So, ΔT1 = final temperature - 95
Step 2: Calculate the heat gained by the water and the calorimeter:
Q2 = (m2 + m3) * c2 * ΔT2
where,
m2 = mass of water = 60g
m3 = mass of calorimeter = 100g
c2 = specific heat capacity of water and calorimeter = 419 J/kg·°C
ΔT2 = change in temperature = final temperature - 20
Step 3: Since the system is thermally isolated, the heat lost by the zinc (Q1) will be equal to the heat gained by the water and the calorimeter (Q2).
Therefore, Q1 = Q2
Setting the equations equal to each other:
m1 * c1 * (final temperature - 95) = (m2 + m3) * c2 * (final temperature - 20)
Since we want to find the final temperature, we can rearrange the equation:
(m1 * c1 * final temperature) - (m1 * c1 * 95) = (m2 + m3) * c2 * final temperature - (m2 + m3) * c2 * 20
Bring the terms with the final temperature to one side:
m1 * c1 * final temperature - (m2 + m3) * c2 * final temperature = (m1 * c1 * 95) - (m2 + m3) * c2 * 20
Common factor out the final temperature:
(final temperature) * ( m1 * c1 - (m2 + m3) * c2) = (m1 * c1 * 95) - (m2 + m3) * c2 * 20
Divide both sides by ( m1 * c1 - (m2 + m3) * c2) to isolate the final temperature:
final temperature = ((m1 * c1 * 95) - (m2 + m3) * c2 * 20) / ( m1 * c1 - (m2 + m3) * c2)
Now, substitute the given values into the equation and solve to find the final temperature.
To find the final temperature, we can use the principle of conservation of energy. The total heat lost by zinc will be equal to the total heat gained by water and the calorimeter.
The heat lost by the zinc can be calculated using the formula:
Q(zinc) = m(zinc) * c(zinc) * ΔT(zinc)
Where:
- m(zinc) is the mass of zinc (100g)
- c(zinc) is the specific heat capacity of zinc (377 J/kg degrees Celsius)
- ΔT(zinc) is the change in temperature of zinc (final temperature - initial temperature of zinc)
The heat gained by the water and calorimeter can be calculated using the formula:
Q(water+calorimeter) = (m(water) + m(calorimeter)) * c(water+calorimeter) * ΔT(water+calorimeter)
Where:
- m(water) is the mass of water (60g)
- m(calorimeter) is the mass of the calorimeter (100g)
- c(water+calorimeter) is the specific heat capacity of water and calorimeter (419 J/kg degrees Celsius)
- ΔT(water+calorimeter) is the change in temperature of water and calorimeter (final temperature - initial temperature of water and calorimeter)
Since the heat lost by the zinc is equal to the heat gained by the water and calorimeter, we can set up an equation:
Q(zinc) = Q(water+calorimeter)
m(zinc) * c(zinc) * ΔT(zinc) = (m(water) + m(calorimeter)) * c(water+calorimeter) * ΔT(water+calorimeter)
Substituting the given values:
100g * 377 J/kg degrees Celsius * ΔT(zinc) = (60g + 100g) * 419 J/kg degrees Celsius * ΔT(water+calorimeter)
Now, we can solve for ΔT(water+calorimeter). Rearranging the equation:
ΔT(water+calorimeter) = (100g * 377 J/kg degrees Celsius * ΔT(zinc)) / ((60g + 100g) * 419 J/kg degrees Celsius)
Now we can substitute the values: ΔT(zinc) = 95°C - 20°C = 75°C
ΔT(water+calorimeter) = (100g * 377 J/kg degrees Celsius * 75°C) / ((60g + 100g) * 419 J/kg degrees Celsius)
Calculating this gives ΔT(water+calorimeter) ≈ 32.92°C
Finally, the final temperature can be calculated by adding ΔT(water+calorimeter) to the initial temperature of the water and calorimeter:
Final temperature = 20°C + 32.92°C = 52.92°C
Therefore, the final temperature will be approximately 52.92 degrees Celsius.