Question

For the following processes, calculate the change in internal energy of the system and determine whether the process is endothermic or exothermic: (a) A balloon is heated by adding 850 J of heat. It expands, doing \(382 \mathrm{~J}\) of work on the atmosphere. (b) A \(50-g\) sample of water is cooled from \(30^{\circ} \mathrm{C}\) to \(15^{\circ} \mathrm{C}\), thereby losing approximately \(3140 \mathrm{~J}\) of heat. (c) A chemical reaction releases \(6.47 \mathrm{~kJ}\) of heat and does no work on the surroundings.

Short answer

(a) The change in internal energy is \(\Delta U = 468 \mathrm{~J}\), and the process is endothermic. (b) The change in internal energy is \(\Delta U = -3140 \mathrm{~J}\), and the process is exothermic. (c) The change in internal energy is \(\Delta U = -6470 \mathrm{~J}\), and the process is exothermic.

Step-by-step solution

Determine the heat added and work done

We're given that the balloon is heated by adding \(850 \mathrm{~J}\) of heat (Q) and it does \(382 \mathrm{~J}\) of work (W) on the atmosphere.

Apply the first law of thermodynamics to calculate ∆U

Now we will plug in the values of Q and W into the equation: ∆U = Q - W ∆U = 850 J - 382 J ∆U = 468 J

Determine if the process is endothermic or exothermic

Since heat is added to the system (Q > 0), the process is endothermic. (b) A \(50-\mathrm{g}\) sample of water is cooled from \(30^{\circ} \mathrm{C}\) to \(15^{\circ} \mathrm{C}\), thereby losing approximately \(3140 \mathrm{~J}\) of heat.

Determine the heat lost

The sample loses \(3140 \mathrm{~J}\) of heat, so Q = -3140 J (since heat is lost, it's a negative value).

Determine the work done

There is no information given about work done in this case, so we can assume W = 0.

Apply the first law of thermodynamics to calculate ∆U

Now we will plug in the values of Q and W into the equation: ∆U = Q - W ∆U = -3140 J - 0 ∆U = -3140 J

Determine if the process is endothermic or exothermic

Since heat is lost from the system (Q < 0), the process is exothermic. (c) A chemical reaction releases \(6.47 \mathrm{~kJ}\) of heat and does no work on the surroundings.

Determine the heat released and work done

The reaction releases \(6.47 \mathrm{~kJ}\) of heat, so Q = -6470 J (since heat is released, it's a negative value). There is no work done on the surroundings, so W = 0.

Apply the first law of thermodynamics to calculate ∆U

Now we will plug in the values of Q and W into the equation: ∆U = Q - W ∆U = -6470 J - 0 ∆U = -6470 J

Determine if the process is endothermic or exothermic

Since heat is released from the system (Q < 0), the process is exothermic.

Key concepts

Internal Energy

Internal energy is a key concept in thermodynamics that refers to the total energy contained within a system. It encompasses all forms of energy that the molecules of a substance possess, which includes kinetic energy due to their motion and potential energy due to the forces between them.

The internal energy of a system changes either by transferring heat to or from the system, or when the system does work, or has work done on it. Importantly, we cannot measure the absolute value of internal energy but can only track the changes in internal energy, which we denote as \( \Delta U \).

When a system's internal energy increases, we say it has absorbed energy, and when the internal energy decreases, it means the system has released energy.

Endothermic Process

An endothermic process is one that absorbs heat from its surroundings. This means that the system requires energy to proceed, and this energy is usually absorbed in the form of heat. As a result, the internal energy of the system increases during such processes. \( \Delta U > 0 \).

For example, when a balloon is heated, it absorbs energy. This energy absorption makes the internal gas molecules more energetic, increasing the internal energy of the balloon and thereby expanding it. This process requires an input of energy, so it is considered endothermic.

Exothermic Process

Conversely, an exothermic process is one that releases heat to its surroundings. The system loses energy during an exothermic reaction, usually in the form of heat, leading to a decrease in its internal energy \( \Delta U < 0 \).

For instance, when a sample of water cools down, it loses heat to the environment. Cooling is an exothermic process because the system's internal energy decreases due to the release of heat.

Heat Transfer

Heat transfer is the movement of energy from one place to another as a result of temperature difference. Heat can be transferred in three ways: conduction, convection, and radiation. In thermodynamics problems, such as those involving the first law, we focus on the quantity of heat transferred, symbolized by \( Q \).

Positive \( Q \) values indicate heat flowing into the system (endothermic process), while negative \( Q \) values represent heat flowing out of the system (exothermic process). The direction of heat transfer affects the system's internal energy and can be directly related to the sensation of heat or coolness in a particular process.

Work Done by System

In thermodynamics, 'work' refers to the energy transferred when a force moves an object over a distance. For a system, work is done when it expands or contracts against an external pressure. We symbolize work done by \( W \), and it is an important aspect of energy exchange.

When a system does work on its surroundings, as in the case of an expanding balloon, the work is considered positive, and the system's internal energy decreases \( \Delta U < 0 \). Conversely, when work is done on the system by its surroundings, the internal energy of the system increases \( \Delta U > 0 \). The first law of thermodynamics precisely accounts for this energy balance involving heat transfer and work done.