Grade 9

Intermediate

9 min

Lesson 8 of 12

Calorimetry and Energy Exchange

Not Started

Using the law of conservation of energy to measure heat transfer between objects.

Have you ever wondered why a small piece of hot metal can boil a thimble of water, but barely warms a bathtub? The secret lies in how energy 'trades places' until everything reaches a perfect balance.

Learning Objectives

1.Apply the principle that heat lost by one object must equal the heat gained by another in a closed system.
2.Identify the components of a calorimeter and explain how they minimize energy loss.
3.Calculate the final equilibrium temperature of a system using the heat transfer equation.

The Law of Heat Exchange

In physics, energy is never lost; it just moves. When a hot object touches a cold one, thermal energy flows from the hotter substance to the cooler one. This continues until they reach thermal equilibrium, meaning they are at the exact same temperature. In an isolated system, we use the Law of Conservation of Energy to state:

Q_{lost} + Q_{gained} = 0

or more simply,

Q_{lost} = -Q_{gained}

. This means every Joule of energy that leaves the hot object must enter the cold one. We calculate this heat (

Q

) using the formula

Q = mc\Delta T

, where

m

is mass,

c

is specific heat capacity, and

\Delta T

is the change in temperature (

T_{final} - T_{initial}

Quick Check

If a hot piece of iron is dropped into cold water and the system is perfectly insulated, what can we say about the total energy change of the iron-water system?

Think about the Law of Conservation of Energy.

Answer

The total energy change is zero because the heat lost by the iron is exactly equal to the heat gained by the water.

The Tool: The Calorimeter

To measure heat transfer accurately, scientists use a calorimeter. A simple version is the 'coffee-cup calorimeter,' made of two nested Styrofoam cups. Why Styrofoam? Because it is an excellent insulator, preventing heat from escaping to the air. A calorimeter typically includes: 1. An insulated container to hold the liquid. 2. A thermometer to track temperature changes. 3. A stirrer to ensure the temperature is uniform throughout the liquid. By keeping the energy 'trapped' inside, we can ensure our measurements of $Q_{lost}$ and $Q_{gained}$ are as precise as possible.

Suppose you mix $0.1\text{ kg}$ of water at $80^{\circ}\text{C}$ with $0.1\text{ kg}$ of water at $20^{\circ}\text{C}$ . What is the final temperature?

1. Since the masses and substances (water) are identical, the heat lost by the hot water will exactly balance the heat gained by the cold water at the midpoint. 2.

T_f = \frac{80 + 20}{2} = 50^{\circ}\text{C}

3. In this simple case, the equilibrium is the simple average because

m_1c_1 = m_2c_2

Solving for Equilibrium

When substances are different, we must use the full equation:

m_1c_1(T_f - T_{i1}) = -m_2c_2(T_f - T_{i2})

. To solve for the final temperature (

T_f

), you must distribute the values and isolate

T_f

. Remember that

\Delta T

is always

(T_{final} - T_{initial})

. For the hotter object,

\Delta T

will be negative (since it cools down), and for the cooler object, it will be positive. The negative sign in the formula accounts for this, ensuring the energy balance is mathematically sound.

A $0.05\text{ kg}$ block of aluminum ( $c = 900\text{ J/kg}^{\circ}\text{C}$ ) at $100^{\circ}\text{C}$ is placed in $0.10\text{ kg}$ of water ( $c = 4186\text{ J/kg}^{\circ}\text{C}$ ) at $20^{\circ}\text{C}$ . Find $T_f$ .

1. Set up the equation: $m_{al}c_{al}(T_f - 100) = -m_w c_w(T_f - 20)$ 2. Plug in values: $(0.05)(900)(T_f - 100) = -(0.10)(4186)(T_f - 20)$ 3. Simplify: $45(T_f - 100) = -418.6(T_f - 20)$ 4. Expand: $45T_f - 4500 = -418.6T_f + 8372$ 5. Group $T_f$ : $463.6T_f = 12872$ 6. Solve: $T_f \approx 27.77^{\circ}\text{C}$

Quick Check

In the equation $Q = mc(T_f - T_i)$ , if an object loses heat, will the value of $Q$ be positive or negative?

Final minus initial.

Answer

Negative, because $T_f$ will be lower than $T_i$ , making the change in temperature negative.

You drop a $0.2\text{ kg}$ mystery metal at $150^{\circ}\text{C}$ into $0.5\text{ kg}$ of water at $20^{\circ}\text{C}$ . The final temperature is $25^{\circ}\text{C}$ . What is the specific heat of the metal?

1. Calculate heat gained by water: $Q_w = (0.5)(4186)(25 - 20) = 10465\text{ J}$ 2. Since $Q_{lost} = -Q_{gained}$ , the metal lost $10465\text{ J}$ . 3. Set up metal equation: $-10465 = (0.2)(c_{metal})(25 - 150)$ 4. Simplify: $-10465 = (0.2)(c_{metal})(-125)$ 5. Solve for $c$ : $-10465 = -25c_{metal} \rightarrow c_{metal} = 418.6\text{ J/kg}^{\circ}\text{C}$

Key Takeaways

1The Law of Heat Exchange states that in an isolated system, heat lost equals heat gained ( $Q_{lost} = -Q_{gained}$ ).
2Thermal equilibrium is the state where two objects reach the same temperature and heat flow stops.
3A calorimeter uses insulation to minimize heat exchange with the environment, allowing for accurate energy measurements.
4The master formula for these problems is $m_1c_1(T_f - T_{i1}) = -m_2c_2(T_f - T_{i2})$ .

Quick Check

Multiple Choice

What is the primary purpose of the Styrofoam in a coffee-cup calorimeter?

Multiple Choice

If $50\text{ J}$ of heat is added to a system, and the system is in thermal equilibrium with its surroundings, what is the net change in energy of the universe?

True/False

Thermal equilibrium is reached when two objects have the same amount of total thermal energy, regardless of their temperature.

What's Next?

Review Tomorrow

In 24 hours, try to write down the heat exchange equation from memory and explain why the negative sign is necessary on one side.

Practice Activity

Find the specific heat capacities of copper, gold, and iron. If you had $1\text{kg}$ of each at $100^{\circ}\text{C}$ , which one would raise the temperature of a bowl of water the most?

Browse

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Calorimetry and Energy Exchange

Learning Objectives

The Law of Heat Exchange

The Tool: The Calorimeter

Solving for Equilibrium

Key Takeaways

Quick Check

What's Next?

Calorimetry and Energy Exchange

Learning Objectives

The Law of Heat Exchange

The Tool: The Calorimeter

Solving for Equilibrium

Key Takeaways

Quick Check

What's Next?