Question

In the equation \(w=-P \Delta V,\) why is there a negative sign?

Short answer

The negative sign in the equation \(w = -P \Delta V\) is essential to maintain the correct sign convention for the work done by the gas during its expansion or contraction. It ensures that when the gas expands (positive \(\Delta V\)), work is done by the system on the surroundings (positive work), and when the gas contracts (negative \(\Delta V\)), work is done on the system by the surroundings (negative work). The negative sign helps conform to the thermodynamic sign convention for work.

Step-by-step solution

Overview of System and Surroundings

In thermodynamics, we typically divide the world into two parts: the system and the surroundings. The system is the part of the world we are studying, such as the gas in a piston. The surroundings include everything else, mainly the environment that is interacting with our system.

Sign Convention for Work

Thermodynamics has a sign convention for work: 1. When the work is done BY the system ON the surroundings, that work is considered positive. An example of this is gas expanding inside a piston, pushing the piston outward and doing work against the external pressure. 2. When the work is done ON the system BY the surroundings, that work is considered negative. An example of this is gas being compressed inside a piston, as the piston pushes inward, working against the pressure of the gas.

Pressure-Volume Work

In the given equation, \(w = -P \Delta V\), we are dealing with a special kind of work called pressure-volume work. It represents the work done due to a change in volume of the gas, under a constant external pressure (P). The term \(\Delta V\) denotes the change in volume and is the final volume (V₂) minus the initial volume (V₁), i.e., \(\Delta V = V₂ - V₁\).

Role of the Negative Sign

Now we understand that when the gas expands (\(V₂ > V₁\)), work is done BY the system ON the surroundings, which should result in positive work. However, when the gas contracts (\(V₂ < V₁\)), work is done ON the system BY the surroundings, which should result in negative work. The negative sign in the equation \(w = -P \Delta V\) ensures that the sign convention holds true. If the gas expands, \(\Delta V\) is positive, and the equation yields a positive value of work since we are multiplying a positive \(\Delta V\) by a negative constant (-P). Conversely, if the gas contracts, \(\Delta V\) is negative, and the equation yields a negative value for work since we are multiplying a negative \(\Delta V\) by a negative constant (-P). Thus, the negative sign in the equation \(w = -P \Delta V\) is essential to maintain the correct sign convention for the work done by the gas during its expansion or contraction.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. The laws of thermodynamics govern the principles of energy transfer and conversion and can be applied to a wide range of processes, including chemical reactions, phase changes, and mechanical movements such as the expansion or compression of gases.

At the heart of thermodynamics lies the concept of a thermodynamic system, an entity or quantity of matter of fixed identity around which we can draw a boundary. Everything outside this boundary is the surroundings. Thermodynamics helps us understand how energy is exchanged and conserved within these boundaries. The concepts of absolute zero, entropy, enthalpies, and free energy are all key to thermodynamics, allowing the prediction of the spontaneous direction of a process and the energy required for reactions and phase transitions.

System and Surroundings

In thermodynamics, it is crucial to define the system and surroundings. The system refers to the specific part of the universe that is being studied or observed, usually enclosed within a physical boundary. This can be as simple as gas within a cylinder or as complex as an entire power plant. The surroundings constitute everything outside that boundary and can interact with the system by exchanging energy in the form of work or heat.

The interaction between the system and its surroundings determines the change in the system’s energy. In a closed system, matter cannot cross the boundary, but energy in the form of heat or work can. By examining these interactions, we gain insights into energy changes and directionality of processes, both critical for the theoretical and practical applications in science and engineering.

Sign Convention for Work

Understanding the sign convention for work in thermodynamics is essential for correctly interpreting the direction of energy transfer between a system and its surroundings. The sign of the work done is determined by the direction of the force and displacement.

Expansion Work

If a system (like gas) expands against the external pressure, it does work on its surroundings. According to the convention, we consider this work positive because the system loses energy. Conversely, when a system is compressed, the surroundings do work on it, which is classified as negative work since the system gains energy.

It’s vital for students to grasp that this convention is a matter of definition and differs from disciplines like engineering, where the opposite sign convention is often used. In thermodynamics, the focus is on the system's perspective.

Work Done by Gas

The concept of work done by a gas is a primary component of applied thermodynamics, especially when dealing with the operation of heat engines, refrigerators, and other systems where gases are used to transfer energy. When a gas expands, it pushes against a force, like the pistons in an engine, and does work on its surroundings.

Calculating Work Done by Gas

The work done by a gas during expansion or compression can be calculated using the formula: \( w = -P \Delta V \), where \( P \) is the constant external pressure, and \( \Delta V \) is the change in volume of the gas. The negative sign in front of the pressure reflects the thermodynamic convention that work done by the system is positive and work done on the system is negative. This equation is fundamental for engineers and scientists to design systems and predict behavior in processes involving gas phase changes.