Thermal Equilibrium: Videos & Practice Problems

Thermal equilibrium occurs when two substances at different temperatures come into physical contact and reach the same temperature, ceasing the exchange of thermal energy according to the first law of thermodynamics. In this state, the heat lost by the hotter object (negative Q) is equal to the heat gained by the colder one (positive Q), as represented by the equation:

$Q=mc\Delta T$

where m is mass, c is specific heat capacity, and ΔT is the change in temperature. If the system is not ideal, the calorimeter or container may also absorb heat, modifying the equation to include the heat absorbed by the calorimeter (Q of the container). This concept is crucial in understanding energy transfer and the approach to thermal balance in physical systems.

Thermal Equilibrium

Thermal equilibrium occurs when two substances in contact reach the same temperature, resulting in no net exchange of thermal energy, as described by the first law of thermodynamics. For instance, consider a heated metal cube at 110 degrees Celsius placed in water at 40 degrees Celsius. Heat naturally transfers from the hotter object (the cube) to the colder one (the water). In this scenario, the cube loses heat, represented as a negative heat transfer (q), while the water gains heat, indicated as a positive q. This process continues until both the cube and the water reach the same temperature, signifying thermal equilibrium.

At thermal equilibrium, the heat lost by the hotter object equals the heat gained by the colder object, expressed mathematically as:

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

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

$$-m_{\text{cube}}c_{\text{cube}}(T_f - T_{\text{cube}}) = m_{\text{water}}c_{\text{water}}(T_f - T_{\text{water}})$$

where \(T_f\) is the final equilibrium temperature.

In ideal conditions, heat transfer occurs solely between the heated object and the solvent. However, in non-ideal situations, the calorimeter (or container) may also absorb some heat. In such cases, the equation expands to account for the calorimeter's heat absorption:

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

Thus, when solving thermal equilibrium problems, if only two substances are involved, the heat lost by the hotter object equals the heat gained by the colder object. If the calorimeter is included, its heat absorption must also be considered to accurately reflect the total heat transfer in the system.

Thermal Equilibrium Example 1

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 the same 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 formula for calculating the final temperature (\(T_f\)) in such a system can be expressed as:

\[ m_{lead} \cdot c_{lead} \cdot (T_{initial, lead} - T_f) = m_{solution} \cdot c_{solution} \cdot (T_f - T_{initial, solution}) \]

Where:

• \(m_{lead}\) = mass of the lead block
• \(c_{lead}\) = specific heat capacity of lead
• \(T_{initial, lead}\) = initial temperature of the lead
• \(m_{solution}\) = mass of the solution
• \(c_{solution}\) = specific heat capacity of the solution
• \(T_{initial, solution}\) = initial temperature of the solution

By applying this equation, one can determine the exact final temperature of the solution after the heat exchange occurs. The key takeaway is that the final temperature will always lie between the initial temperatures of the two substances involved in the heat transfer.

If 53.2 g Al at 120.0 ºC is placed in 110.0 g H2O at 90 ºC within an insulated container that absorbs a negligible amount of heat, what is the final temperature of the aluminum? The specific heat capacities of water and aluminum are 4.184 J/g ∙ ºC and 0.897 J/g ∙ ºC, respectively.

• A. 75.579 °C
• B. 92.775 °C
• C. 72.975 °C
• D. 57.972 °C

Answer: (B)

Here’s what students ask on this topic:

What is thermal equilibrium?

Thermal equilibrium is a concept in thermodynamics that describes a state where two objects or systems in physical contact with each other exchange heat energy until they reach the same temperature. At this point, there is no net flow of heat between the objects; the amount of heat leaving one object is equal to the amount of heat entering it from the other object. When thermal equilibrium is achieved, the systems are said to be at the same temperature and there is no further change in their thermal properties. This is a fundamental principle underlying the zeroth law of thermodynamics, which states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This principle allows us to use thermometers to measure temperature objectively.

What does it mean when we say that an object is in thermal equilibrium?

When an object is in thermal equilibrium, it means that it has reached a state where there is no net exchange of heat energy with its surroundings. This occurs when the object and its environment are at the same temperature, so heat, which naturally flows from hotter to cooler regions, no longer transfers between them. In this state, all parts of the object and its immediate environment are at a uniform temperature. Thermal equilibrium is a key concept in thermodynamics, and it's essential for understanding how heat transfer works. It's also the principle behind thermometers, which measure their own temperature change until they reach thermal equilibrium with the object or environment they are measuring, thus indicating the temperature.

How do you calculate the temperature of thermal equilibrium?

To calculate the temperature of thermal equilibrium, you need to consider the principle that when two objects of different temperatures are in thermal contact, heat energy will transfer from the hotter object to the cooler one until they reach the same temperature. This final temperature is the temperature of thermal equilibrium.

If you are dealing with two substances, you can use the formula:

$\left(Q_{\text{lost}}\right)+\left(Q_{\text{gained}}\right)=0$

where Q represents the heat energy lost or gained, which can be calculated using the specific heat capacity equation:

$Q=mc\Delta T$

Here, 'm' is the mass, 'c' is the specific heat capacity of the substance, and 'ΔT' is the change in temperature.

For two objects, the equation becomes:

$\left(m_1\right)\left(c_1\right)(T_f-T_1)+\left(m_2\right)\left(c_2\right)(T_f-T_2)=0$

Solve for Tf, which is the final temperature of thermal equilibrium. You'll need to know the initial temperatures (T1 and T2), masses (m1 and m2), and specific heat capacities (c1 and c2) of the two substances involved.