Question

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

Short answer

The negative sign in the equation \(w = -P \Delta V\) holds physical significance to indicate the direction of work done and the associated transfer of energy during a gas's volume change. When the gas expands (\(\Delta V > 0\)), it performs work on its surroundings, and the system loses energy, resulting in a negative value for work. Conversely, when the gas is compressed (\(\Delta V < 0\)), work is done on the gas, transferring energy into the system, yielding a positive value for work. Without the negative sign, the relationship between work, pressure, and volume change in thermodynamics would be misrepresented.

Step-by-step solution

Concepts Involved

In thermodynamics, we study the work done by or on a gas when it undergoes a change in volume under the influence of pressure. The equation given is \(w = -P \Delta V\), where \(w\) represents the work done, \(P\) represents the pressure, and \(\Delta V\), the change in volume.

Significance of Negative Sign

The negative sign in the equation \(w = -P \Delta V\) has a physical significance relating to the direction of work done. Work is said to be done on a system when energy is transferred into the system, and work is said to be done by the system when energy is transferred out of the system. In the context of this equation, a positive change in volume (\(\Delta V > 0\)) represents an expansion of the gas, indicating that work is done by the gas to push its surroundings outward. In this case, the gas is doing the work, and the system loses energy. As a result, we assign a negative sign to the work done in this case. On the other hand, a negative change in volume (\(\Delta V < 0\)) represents a compression of the gas, indicating that work is done on the gas by the surroundings. In this case, energy is transferred into the system, and we assign a positive value to the work done in this case.

Concluding the Significance

The negative sign in the equation \(w = -P \Delta V\) is essential to indicate the direction of work done and the associated transfer of energy. A negative sign signifies that the gas is transferring energy out of the system (work done by the gas), and a positive sign signifies that the gas is receiving energy from the surroundings (work done on the gas). Without the negative sign, the equation would not correctly represent the relationship between work, pressure, and volume change in thermodynamics.