51.59 g of water at 82.6 oC is added to a calorimeter that contains 48.01 g of water at 40.9 oC. If the final temperature of the system is 59.4 oC, what is the calorimeter constant (Ccalorimeter) ? Use 4.184 J/goC for the heat capacity of water.
heat gained by cool water + heat lost by warm water + heat lost by calorimeter = 0
[mass cool water x specific heat x (Tfinal-Tinitial)] + [mass warm water x specific heat x (Tfinal-Tinitial)] + Ccal x (Tfinal-Tinitial) = 0
Substitute FIRST, then solve for Ccal.
72.97
69.82
To find the calorimeter constant (Ccalorimeter), we need to use the principle of heat transfer. The heat lost by the hot water is equal to the heat gained by the cold water and the calorimeter.
The heat gained or lost by a substance can be calculated using the formula:
Q = m * C * ΔT
Where:
Q is the heat gained or lost (in joules or calories),
m is the mass of the substance (in grams),
C is the specific heat capacity of the substance (in J/goC or cal/goC),
ΔT is the change in temperature (in oC).
First, let's calculate the heat lost by the hot water:
Q_hot = m_hot * C_water * ΔT_hot
Where:
m_hot is the mass of the hot water (51.59 g),
C_water is the specific heat capacity of water (4.184 J/goC),
ΔT_hot is the change in temperature for the hot water (final temperature - initial temperature).
ΔT_hot = 59.4 oC - 82.6 oC = -23.2 oC
Q_hot = 51.59 g * 4.184 J/goC * -23.2 oC
Next, let's calculate the heat gained by the cold water and the calorimeter:
Q_cold = (m_cold + m_calorimeter) * C_water * ΔT_cold
Where:
m_cold is the mass of the cold water (48.01 g),
m_calorimeter is the mass of the calorimeter (we'll call it m_calorimeter),
ΔT_cold is the change in temperature for the cold water and the calorimeter (final temperature - initial temperature).
ΔT_cold = 59.4 oC - 40.9 oC = 18.5 oC
Q_cold = (48.01 g + m_calorimeter) * 4.184 J/goC * 18.5 oC
Since the heat lost by the hot water (Q_hot) is equal to the heat gained by the cold water and the calorimeter (Q_cold), we can set up an equation:
Q_hot = Q_cold
51.59 g * 4.184 J/goC * -23.2 oC = (48.01 g + m_calorimeter) * 4.184 J/goC * 18.5 oC
Simplifying the equation:
-52.52 g * oC = (48.01 g + m_calorimeter) * 18.5 oC
-52.52 g * oC = 888.185 oC * g + 18.5 oC * m_calorimeter
Now, we need to isolate m_calorimeter:
-52.52 g * oC - 888.185 oC * g = 18.5 oC * m_calorimeter
-940.705 g * oC = 18.5 oC * m_calorimeter
m_calorimeter = -940.705 g * oC / 18.5 oC
m_calorimeter ≈ -50.867 g
Since the mass cannot be negative, we can ignore the negative sign. Therefore, the mass of the calorimeter is approximately 50.867 g.
The calorimeter constant (Ccalorimeter) is defined as the heat capacity of the calorimeter divided by its mass:
Ccalorimeter = Q_cold / (m_calorimeter * ΔT_cold)
Substituting the values we have:
Ccalorimeter ≈ (48.01 g + 50.867 g) * 4.184 J/goC * 18.5 oC / (50.867 g * 18.5 oC)
Simplifying the equation:
Ccalorimeter ≈ 98.8774 J/oC
Therefore, the calorimeter constant (Ccalorimeter) is approximately 98.8774 J/oC.