A 15.0g sample of quartz, which has a specific heat capacity of 0.730·J·g−1°C−1, is dropped into an insulated container containing 150.0g of water at 55.0°C and a constant pressure of 1atm. The initial temperature of the quartz is 8.0°C.Assuming no heat is absorbed from or by the container, or the surroundings, calculate the equilibrium temperature of the water. Be sure your answer has the correct number of significant digits.°C
Question
A 15.0g sample of quartz, which has a specific heat capacity of 0.730·J·g−1°C−1, is dropped into an insulated container containing 150.0g of water at 55.0°C and a constant pressure of 1atm. The initial temperature of the quartz is 8.0°C.Assuming no heat is absorbed from or by the container, or the surroundings, calculate the equilibrium temperature of the water. Be sure your answer has the correct number of significant digits.°C
Solution
To solve this problem, we need to use the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred. In this case, the heat lost by the water will be equal to the heat gained by the quartz.
The formula for heat transfer is Q = mcΔT, where:
- Q is the heat transferred,
- m is the mass,
- c is the specific heat capacity, and
- ΔT is the change in temperature.
For the water, the heat lost (Q_water) can be calculated as:
Q_water = m_water * c_water * (T_initial_water - T_final)
For the quartz, the heat gained (Q_quartz) can be calculated as:
Q_quartz = m_quartz * c_quartz * (T_final - T_initial_quartz)
Since the heat lost by the water is equal to the heat gained by the quartz, we can set these two equations equal to each other and solve for the final temperature (T_final):
m_water * c_water * (T_initial_water - T_final) = m_quartz * c_quartz * (T_final - T_initial_quartz)
Substituting the given values:
150.0g * 4.18 J/g°C * (55.0°C - T_final) = 15.0g * 0.730 J/g°C * (T_final - 8.0°C)
Solving this equation for T_final will give us the equilibrium temperature of the water.
Let's simplify the equation:
627.0°C * (55.0°C - T_final) = 10.95°C * (T_final - 8.0°C)
34585°C - 627.0°C * T_final = 10.95°C * T_final - 87.6°C
34585°C + 87.6°C = 627.0°C * T_final + 10.95°C * T_final
34672.6°C = 637.95°C * T_final
T_final = 34672.6°C / 637.95°C = 54.3°C
So, the equilibrium temperature of the water is 54.3°C.
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