4.3.2 Internal Energy and Energy Transfers

In our study of the particle model of matter, we have seen that substances are made of atoms and molecules that are constantly in motion. Because these particles move and interact, they possess energy. This total energy stored within a system is known as Internal Energy.

4.3.2.1 Internal Energy

Definition

Internal energy is the total energy stored by the particles (atoms and molecules) that make up a system.

It is the sum of two specific energy stores:

1. Kinetic Energy Store: Due to the motion of the particles (vibration, rotation, or translation).
2. Potential Energy Store: Due to the positions of the particles relative to each other (the bonds/intermolecular forces between them).

The Formula

The internal energy of a system can be expressed as:

$U=E_k+E_p$

Where:

1. U = Internal Energy (Joules, J)
2. $E_k$ = Total Kinetic Energy of the particles (Joules, J)
3. $E_p$ = Total Potential Energy of the particles (Joules, J)

Explanation: How Heating Affects Internal Energy

When you heat a system, you transfer energy to its particles. This energy transfer increases the internal energy of the system. This increase in internal energy results in one of two things:

1. A Change in Temperature: The kinetic energy of the particles increases. Since temperature is a measure of the average kinetic energy of the particles, the object gets hotter.
2. A Change in State: The potential energy of the particles increases. The energy is used to break or weaken the bonds between particles rather than making them move faster. In this case, the temperature stays constant during the change of state (e.g., melting or boiling).

Factors Affecting Internal Energy

1. Mass: A larger mass of the same substance at the same temperature will have more internal energy because there are more particles.
2. Temperature: Increasing the temperature increases the kinetic energy of the particles, thus increasing internal energy.
3. Material: Different materials store different amounts of energy based on their molecular structure and bond strength.

Worked Example: Internal Energy Concepts

Question: A block of ice at $0^{\circ }\text{C}$ is heated until it becomes liquid water at $0^{\circ }\text{C}$. Describe what happens to the internal energy, kinetic energy, and potential energy of the particles during this process.

Answer:

1. Step 1: Identify the change. The substance is changing state from solid to liquid at a constant temperature.
2. Step 2: Evaluate Kinetic Energy ($E_k$). Since the temperature remains constant ($0^{\circ }\text{C}$), the average kinetic energy of the particles does not change.
3. Step 3: Evaluate Potential Energy ($E_p$). Energy is being absorbed to break the bonds of the solid lattice. Therefore, the potential energy of the particles increases.
4. Step 4: Conclude Internal Energy (U). Since $U=E_k+E_p$, and $E_p$ has increased while $E_k$ stayed the same, the total internal energy of the system has increased.

Edge Cases and Common Misconceptions

1. "Internal" vs. "External" Energy: A ball flying through the air has kinetic energy because of its overall motion, but this is not part of its internal energy. Internal energy only refers to the energy of the particles inside the ball relative to each other.
2. Absolute Zero: At 0 Kelvin (Absolute Zero), the kinetic energy of particles is at its absolute minimum (theoretically zero). However, substances still possess some internal energy due to their potential energy stores.
3. Temperature ≠ Heat: Heat is the transfer of energy. Temperature is a measure of the average kinetic energy. Internal energy is the total sum of all energy stores in the particles.

4.3.2.2 Temperature changes in a system and specific heat capacity

When energy is transferred to a substance, its internal energy increases. If this energy transfer does not cause a change of state (like melting or boiling), the temperature of the substance will rise. However, not all materials heat up at the same rate.

Definition

Specific Heat Capacity (SHC) is the amount of energy required to raise the temperature of one kilogram of a substance by one degree Celsius (or Kelvin).

In simpler terms, it tells us how "difficult" it is to change the temperature of a specific material. For example, water has a very high specific heat capacity, meaning it takes a lot of energy to heat up but also stays warm for a long time.

The Formula

The amount of energy needed to change the temperature of a system can be calculated using the following equation:

$\Delta E=mc\Delta \theta$

Where:

1. $\Delta E$ = Change in thermal energy (Joules, J)
2. m = Mass (Kilograms, kg)
3. c = Specific heat capacity ($J/k{g}^{\circ }\text{C}$)
4. $\Delta \theta$ = Temperature change (${}^{\circ }\text{C}$)

Note: The symbol $\Delta$ (delta) means "change in".

Units

The unit for Specific Heat Capacity is Joules per kilogram per degree Celsius ($J/k{g}^{\circ }\text{C}$).

Key Factors

The increase in temperature of a system depends on three main factors:

1. The mass of the substance: The more particles there are, the more energy is needed to get them all moving faster.
2. The type of material: This is represented by the specific heat capacity (c).
3. The amount of energy put in: More energy results in a higher temperature rise.

Worked Example: Calculating Energy Transfer

Question: Calculate the energy required to increase the temperature of $2kg$ of water from $20^{\circ }\text{C}$ to $100^{\circ }\text{C}$. The specific heat capacity of water is $4,200J/k{g}^{\circ }\text{C}$.

Solution:

1. Step 1: List the known values.
2. $m=2kg$
3. $c=4,200J/k{g}^{\circ }\text{C}$
4. $\Delta \theta =100-20=80^{\circ }\text{C}$
5. Step 2: State the formula.

$\Delta E=mc\Delta \theta$

1. Step 3: Substitute the values.

$\Delta E=2\times 4,200\times 80$

1. Step 4: Calculate the final answer.

$\Delta E=672,000J\text{ (or }672kJ\text{)}$

Edge Cases and Important Tips

1. Rearranging the Formula: You may be asked to find the mass or the specific heat capacity.
2. To find c: $c=\frac{\Delta E}{m\Delta \theta }$
3. Cooling Down: The formula works for cooling too! If a substance cools down, $\Delta E$ will be negative, indicating that energy is being released to the surroundings.
4. Required Practical: In the AQA syllabus, you must know how to determine the specific heat capacity of a material (usually a metal block) experimentally. This involves using an immersion heater, a thermometer, and a joulemeter.
5. Energy Losses: In real-world experiments, the calculated value of c is often higher than the actual value because some thermal energy escapes to the surroundings rather than heating the block.

4.3.2.3 Changes of State and Specific Latent Heat

In the previous section, we learned that heating a substance usually increases its temperature. However, there are specific moments where you can add energy to a substance and its temperature will not change. This happens during a change of state.

Definition

Specific Latent Heat is the amount of energy required to change the state of one kilogram of a substance with no change in temperature.

The word "latent" means "hidden." It is called this because if you were watching a thermometer, the energy being added would seem to "disappear" as the temperature gauge stops rising.

The Two Types of Latent Heat

There are two distinct types of specific latent heat depending on which state change is occurring:

1. Specific Latent Heat of Fusion ($L_f$): The energy needed to change 1kg of a substance from solid to liquid (melting) at its melting point.
2. Specific Latent Heat of Vaporisation ($L_v$ ): The energy needed to change 1kg of a substance from liquid to vapour (boiling) at its boiling point.

The Formula

To calculate the energy required for a change of state, use:

E = mL

Where:

1. E = Thermal energy for a change of state (Joules, J)
2. m = Mass (Kilograms, kg)
3. L = Specific Latent Heat (J/kg)

Explanation: What is happening to the particles?

When a substance changes state:

1. Energy is NOT increasing Kinetic Energy: Therefore, the temperature stays the same.
2. Energy IS increasing Potential Energy: The energy is being used to break the bonds (intermolecular forces) holding the particles together.

Conversely, when a substance condenses or freezes, the particles form new bonds, and energy is released to the surroundings, even though the temperature remains constant.

Heating and Cooling Curves

A heating curve is a graph of temperature against time. You must be able to identify the state changes on these graphs:

1. Sloped lines: The internal energy is increasing because the kinetic energy of particles is increasing (temperature rise).
2. Flat (horizontal) lines: The internal energy is increasing because the potential energy is increasing (change of state).

Worked Example: Latent Heat of Fusion

Question: How much energy is required to melt 0.5kg of ice at $0^{\circ }\text{C}$? The specific latent heat of fusion for water is $334,000J/kg$.

Solution:

1. Step 1: List the known values.
2. $m=0.5kg$
3. $L_f=334,000J/kg$
4. Step 2: State the formula. E = mL
5. Step 3: Substitute the values. $E=0.5\times 334,000$
6. Step 4: Calculate the final answer. $E=167,000J\text{ (or }167kJ\text{)}$

Edge Cases and Common Misconceptions

1. Evaporation vs. Boiling: Evaporation happens at the surface of a liquid at any temperature. Boiling happens throughout the liquid only at the boiling point. Specific Latent Heat calculations specifically refer to boiling.
2. Vaporisation vs. Fusion Values: For most substances, the latent heat of vaporisation is much higher than the latent heat of fusion. This is because it takes much more energy to completely separate particles into a gas than it does to simply loosen them into a liquid.
3. Direction of Energy: The formula E=mL works both ways. If 1kg of steam condenses into water, it releases exactly the same amount of energy ($L_v$) that was required to boil it.