Ideally, when a thermometer is used to measure the temperature of an object, the temperature of the object itself should not change. However, if a significant amount of heat flows from the object to the thermometer, the temperature will change. A thermometer has a mass of 31.8 g, a specific heat capacity of c = 825 J/(kg C°), and a temperature of 12.3 °C. It is immersed in 131 g of water, and the final temperature of the water and thermometer is 60.1 °C. What was the temperature of the water in degrees Celsius before the insertion of the thermometer?
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To solve this problem, we can apply the principle of heat transfer and use the equation:
Q = mcΔT
Where:
Q is the heat transferred (in this case, between the water and thermometer)
m is the mass of the substance (either water or thermometer)
c is the specific heat capacity of the substance
ΔT is the change in temperature
Let's break down the problem and assign variables:
1. Mass of the thermometer: m_thermometer = 31.8 g
2. Specific heat capacity of the thermometer: c_thermometer = 825 J/(kg °C)
3. Temperature of the thermometer: T_thermometer = 12.3 °C
4. Mass of the water: m_water = 131 g
5. Initial temperature of the water: T_initial_water (unknown)
6. Final temperature of the water and thermometer: T_final = 60.1 °C
Now, let's calculate the heat transferred between the water and thermometer:
Q_water = mcΔT
The heat transferred from the water is equal to the heat gained by the thermometer. Assuming no heat is exchanged with the surroundings:
Q_water = -Q_thermometer
Now let's find the specific heat capacity of the water:
c_water = 4186 J/(kg °C) (specific heat capacity of water)
Using these values, we can set up the equation:
m_water * c_water * (T_final - T_initial_water) = -m_thermometer * c_thermometer * (T_final - T_thermometer)
Let's plug in the values:
131 g * 4186 J/(kg °C) * (60.1 °C - T_initial_water) = -31.8 g * 825 J/(kg °C) * (60.1 °C - 12.3 °C)
Now, let's solve for T_initial_water:
(131 g * 4186 J/(kg °C) * (60.1 °C - T_initial_water)) / (-31.8 g * 825 J/(kg °C)) = (59.12 °C)
Simplify the equation further:
(51,817.986 J * (60.1 °C - T_initial_water)) / (-26,235 J) = 59.12 °C
Multiply through by -26,235 J:
(51,817.986 J * (60.1 °C - T_initial_water)) = (-26,235 J * 59.12 °C)
Divide both sides by 51,817.986 J:
60.1 °C - T_initial_water = (-26,235 J * 59.12 °C) / 51,817.986 J
Simplify further:
60.1 °C - T_initial_water = -30.142162 °C
Add T_initial_water to both sides:
60.1 °C + 30.142162 °C = T_initial_water
Solve for T_initial_water:
T_initial_water = 90.242162 °C
Therefore, the initial temperature of the water before the insertion of the thermometer was approximately 90.242162 °C.