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 transfer will take place. Heat naturally flows from the hotter object to the colder one, leading to the cooling of the hot object as it transfers its excess heat to the water.

In this scenario, the object losing heat is assigned a negative value for heat transfer (q), while the water gaining heat is assigned a positive value. The principle of conservation of energy dictates that the heat lost by the hot object equals the heat gained by the water, expressed mathematically as:

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

In terms of specific heat capacity, this relationship can be represented as:

$$-m_{\text{object}} c_{\text{object}} \Delta T_{\text{object}} = m_{\text{water}} c_{\text{water}} \Delta T_{\text{water}}$$

Here, \(m\) represents mass, \(c\) represents specific heat capacity, and \(\Delta T\) represents the change in temperature. 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.

It is crucial to remember that the hotter object will always have a negative q value due to heat loss, while the colder object will have a positive q value as it gains heat. This understanding is essential for correctly applying the principles of thermal equilibrium in various scenarios.

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 = mcΔT

Where:

• Q is the heat transferred (in joules),
• m is the mass (in grams),
• c is the specific heat capacity (in J/g°C),
• ΔT is the change in temperature (in °C).

By applying this formula to both the lead and the solution, we can set up an equation to find the final temperature:

m_lead * c_lead * (T_{initial, lead} - T_final) = m_solution * c_solution * (T_final - T_{initial, solution})

In this equation, the mass and specific heat capacities of both substances will influence the final temperature. Ultimately, the final temperature will be a value that reflects the balance of heat lost and gained, confirming that it lies between the initial temperatures of 90 degrees Celsius and 250 degrees Celsius.