Question

(a) Write an equation that expresses the first law of thermodynamics in terms of heat and work. (b) Under what conditions will the quantities $q$ and $w$ be negative numbers?

Short answer

(a) The first law of thermodynamics equation is $\mathrm{\Delta }U=q-w$. (b) The conditions for negative $q$ and $w$ are: - $q<0$: Heat is released (exothermic process) - $w<0$: Compression work (work done on the system)

Step-by-step solution

(a) Write the first law of thermodynamics equation

To write the equation that represents the first law of thermodynamics, we will use the standard symbols: $\mathrm{\Delta }U$ for the change in internal energy, $q$ for the heat added to the system, and $w$ for the work done by the system. The equation is: $$\mathrm{\Delta }U=q-w$$

(b) Determine the conditions for negative $q$ and $w$

In the equation $\mathrm{\Delta }U=q-w$, $q$ will be negative when heat is released by the system, and $w$ will be negative when work is done on the system. 1. Heat ($q$) is negative when energy is being transferred from the system to the surroundings, which means the system is losing heat to the surroundings. This is called the exothermic process. 2. Work ($w$) is negative when the surroundings exert a force on the system, causing the system to compress or the external pressure to do work on the system. This is called compression work. In summary, the conditions for negative $q$ and $w$ are: - $q<0$: Heat is released (exothermic process) - $w<0$: Compression work (work done on the system)

Key concepts

Understanding Internal Energy

Internal energy, denoted as $\mathrm{\Delta }U$, is a core concept when discussing the first law of thermodynamics. It represents the total energy contained within a system. This energy is the sum of all kinetic and potential energy of particles within the system.

When we talk about change in internal energy, we refer to when the internal energy increases or decreases as a result of heat transfer and work done. If the internal energy of a system increases, it means energy is added; if it decreases, energy is lost.

Internal energy change is dependent on two main factors:

• Heat transfer ($q$): Energy added or removed as heat.
• Work done ($w$): Energy change due to work done on or by the system.

The first law of thermodynamics is often expressed with the equation:$$\mathrm{\Delta }U=q-w$$This equation tells us that the change in internal energy ($\mathrm{\Delta }U$) equals the heat added to the system minus the work done by the system.

Exploring Heat Transfer

Heat transfer, represented by $q$, is the process of energy being transferred between a system and its surroundings due to a temperature difference. This is a critical aspect of the first law of thermodynamics since it directly impacts internal energy changes.

There are two primary scenarios for heat transfer:

• If $q>0$, heat is absorbed by the system from its surroundings. This process is known as endothermic. As a result, the system's internal energy increases.
• If $q<0$, heat is released from the system to its surroundings. Such a process is called exothermic, which leads to a decrease in the system's internal energy.

Heat transfer is a spontaneous process, flowing from a hotter body to a cooler one. In thermodynamic calculations, understanding whether a process is endothermic or exothermic is essential, as it influences how $q$ is applied in the $\mathrm{\Delta }U=q-w$ equation.

The Role of Work Done

Work done, denoted as $w$, plays a crucial role in thermodynamics as it affects the energy of a system. In the context of the first law of thermodynamics, work refers to energy transferred when a force is applied over a distance or when the system expands or contracts relative to its surroundings.

The sign of work done is critical:

• If $w>0$, work is done by the system on the surroundings. This could be the system expanding against an external pressure, leading to decreased internal energy.
• If $w<0$, work is done on the system by the surroundings. For example, if the system is compressed, the system's internal energy increases.

Visualizing these work processes is crucial for understanding energy exchange. When calculating changes in internal energy, considering both heat transfer and work done is vital, guided by the equation:$$\mathrm{\Delta }U=q-w$$This equation ensures we understand whether the system gains or loses energy based on specific conditions.