Thermal Equilibrium (Simplified): Videos & Practice Problems

Thermal equilibrium occurs when two substances in physical contact reach the same temperature, resulting in no net exchange of thermal energy. For instance, if a hot object at 110 degrees Celsius is placed in water at 40 degrees Celsius, heat will transfer from the hotter object to the colder water. This process is governed by the principle that heat flows from a hotter object to a cooler one.

In this scenario, the hot object loses heat, which is represented by a negative value for heat transfer (q), while the water gains heat, indicated by a positive value for q. The relationship between the heat lost by the object and the heat gained by the water can be expressed as:

$$q_{\text{object}} + q_{\text{water}} = 0$$

This implies that:

$$q_{\text{object}} = -q_{\text{water}}$$

Using the specific heat formula, we can express this as:

$$-m_{\text{object}}c_{\text{object}}(T_f - T_{\text{initial, object}}) = m_{\text{water}}c_{\text{water}}(T_f - T_{\text{initial, water}})$$

Here, \(m\) represents mass, \(c\) is the specific heat capacity, and \(T_f\) is the final temperature at thermal equilibrium. The negative sign indicates that the object is losing heat, while the water is gaining heat. Under ideal conditions, heat transfer occurs solely between the heated object and the water, without any loss to the surrounding environment.

In summary, understanding the signs of heat transfer is crucial: the hotter object will have a negative q, while the colder object will have a positive q. This principle is fundamental in thermodynamics and helps predict the behavior of substances in thermal contact.

In this scenario, we are examining the heat transfer between a hot block of lead and a cooler solution. When a hot object, such as a 50-gram block of lead at 250 degrees Celsius, is submerged in a cooler solution at 90 degrees Celsius, heat will flow from the lead to the solution until thermal equilibrium is reached. This process is similar to placing a hot pan into cold water, where the pan releases heat, causing the water to heat up and potentially vaporize.

At thermal equilibrium, both the lead and the solution will reach a common final temperature, which will be somewhere between their initial temperatures. The final temperature can be predicted to be greater than 90 degrees Celsius but less than 250 degrees Celsius. This is because the hotter object (the lead) loses heat while the cooler object (the solution) gains heat. The specific final temperature can be calculated using the principle of conservation of energy, which states that the heat lost by the lead will equal the heat gained by the solution.

The heat transfer can be expressed mathematically using the formula:

\[ Q = mc\Delta T \]

Where:

• Q is the heat transferred (in joules),
• m is the mass (in grams),
• c is the specific heat capacity (in J/g°C),
• \Delta T is the change in temperature (final temperature - initial temperature).

By applying this formula to both the lead and the solution, we can set up an equation to solve for the final temperature, ensuring that the heat lost by the lead equals the heat gained by the solution. This approach reinforces the concept of energy conservation in thermal processes.