Physics 8 Lesson 25: Heat balance equation
1. Theoretical summary
1.1. Principle of heat transfer

Heat transfers from an object with a higher temperature to an object with a lower temperature.

Heat transfer occurs until the temperatures of the two objects are equal.

The heat released by one object is equal to the heat absorbed by the other.
Eg: Dropping a heated metal ingot into a water bath, the initial temperature of the metal ingot is higher than the temperature of the water, so there is a heat exchange: The metal rod gives off heat and decreases the temperature while the water collects heat to increase. temperature. When the temperature of the metal rod and the water are equal, the heat transfer process ends.
1.2. Heat balance equation
Formula: Q_{radiate} = Q_{collect}
Where: \(Q=mC\Delta t\)
\(\Delta t=t_2t_1\)
QResult = m1.C1.(t1 – t2)
Qthu = m2.C2.(t2 – t1)
⇒ \(m_1.C_1.(t_1t)=m_2.C_2.(tt_2)\)
2. Illustrated exercise
2.1. Form 1: Find the specific heat capacity of the metal
To determine the specific heat of a metal, a calorimeter contains 500 g of water at a temperature of 13 .^{0}C a piece of metal of mass 400 g is heated to 100^{0}C. Temperature at thermal equilibrium is 20^{0}C. Calculate the specific heat of the metal. Neglect the calorific value of heating the calorimeter and the air. Take the specific heat capacity of water to be 4190 J/kg.K
Solution guide:
\(Q_1= m_1.c_1.(t_1t_2)= 0.4.c.(10020)\)
\(Q_2= m_2.c_2.(tt_2)= 0.5.4190.(2013)\)
\(Q_{collect}= Q_{prescription} \Rightarrow\) \(0.4.c.(10020)\)\(=0,5.4190.(2013)\)
c = 458 J/kg.K
So, this metal is steel.
2.2. Form 2: Determine the amount of heat received by the water
A piece of copper of mass 0.5 kg is dropped into 500 g of water. Copper plate cools from 80^{0}C down 20^{0}C. How much heat does water receive and by how many degrees does it heat up?
Solution guide:
The heat received by the water is equal to the heat given off by the copper plate:
\(Q=m_1.C_1.(t_1t_2)=0.5.380.(8020)=11400(J)\)
Hot water added up:
t = Q/m_{2}.c_{2}
⇒ ∆t = \(\frac{11 400}{0.5.4 200}=5.43^oC\)
Heat received by water is equal to heat: \(Q= 11400(J)\).
Hot water adds up: ∆t = \(5.43^oC\).
3. Practice
3.1. Essay exercises
Question 1: A piece of copper of mass 0.5 kg is dropped into 500 g of water. The copper plate cools from 80°C to 20°C. How many degrees warmer is the water? The specific heat capacity of copper is 380 J/kg.K and that of water is 4200 J/kg.K.
Verse 2: Mix three chemically inactive liquids of mass m . respectively_{first} = 2 kg, m_{2 }= 3 kg, m_{3} = 4 kg. Knowing their specific heat and temperature are c ., respectively_{first} = 2000 J/kg.K, t_{first} = 57°C, c_{2} = 4000 J/kg.K, t_{2} = 63°C, c_{3} = 3000 J/kg.K, t_{3} = 92°C. What is the temperature of the mixture at equilibrium?
Question 3: Mixing alcohol with water results in a mixture weighing 120.8 g at t = 30°C. Calculate the mass of water and alcohol mixed, given that the alcohol initially has a temperature t_{first} = 10°C and the water has a temperature of t_{2} = 90°C. The specific heat capacity of alcohol and water is c ., respectively_{first} = 2500 J/kg.K, c2 = 4200 J/kg.K.
Question 4: Drop a 0.15 kg aluminum ball heated to 100°C into a beaker of water at 20°C. After some time, the temperature of the sphere and of the water are both 25°C. Consider the sphere and the water to only transfer heat to each other. The specific heat capacity of aluminum and water is 800 J/kg.K, 4200 J/kg.K. What is the mass of water?
3.2. Multiple choice exercises
Question 1: Pour 5 liters of water at 20°C into 3 liters of water at 45°C. The equilibrium temperature is:
A. 2.94°C B. 293.75°C C. 29.36°C D. 29.4°C
Verse 2: Which of the following is true about the principle of heat transfer:
A. Heat transfers itself from an object with a lower temperature to an object with a higher temperature.
B. Heat transfers itself from an object with a higher temperature to an object with a lower temperature.
C. Heat transfers from an object with a higher specific heat to an object with a lower specific heat.
D. Heat transfers from an object with a lower specific heat to an object with a higher specific heat.
Question 3: Drop a 2 kg piece of steel at 345°C into a 3 liter water tank. After equilibration the final temperature is 30°C. Neglect heat loss through the medium. Knowing the specific heat capacity of steel and water are 460 J/kg.K, 4200 J/kg.K, respectively. The initial temperature of the water is:
A. 7°C B. 17°C C. 27°C D. 37°C
Question 4: People want to mix bath water with a temperature of 38°C. How many liters of boiling water must be added to 15 liters of cold water at 24°C?
A. 2.5 liters B. 3.38 liters C. 4.2 liters D. 5 liters
4. Conclusion
Through this lecture on Heat Equilibrium Equations, students need to complete some of the objectives given by the lesson, such as:

State 3 contents of the principle of heat transfer.

Write the heat balance equation for the case where two objects exchange heat with each other.

Solve simple problems involving heat exchange between two objects. Apply the formula for calculating heat.
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