Question

Consider the following cyclic process carried out in two steps on a gas: Step \(1: 45 \mathrm{J}\) of heat is added to the gas, and \(10 . \mathrm{J}\) of expansion work is performed. Step \(2: 60. \mathrm{J}\) of heat is removed from the gas as the gas is compressed back to the initial state. Calculate the work for the gas compression in Step 2.

Short answer

The work done on the gas during the compression in Step 2 is -5 J (negative since it is work done on the gas).

Step-by-step solution

Identify the given information

In this exercise, we are given: - Heat input in Step 1 (Q1) is 45 J - Expansion work in Step 1 (W1) is -10 J (negative since it is work done by the gas) - Heat removed in Step 2 (Q2) is -60 J (negative since it is heat removed from the gas) - Work in Step 2 (W2) needs to be calculated

Apply the cyclic process property

In a cyclic process, the net work done is equal to the net heat transfer, that is: \(W_{net} = Q_{net}\) The net work done is the sum of work done in Step 1 and Step 2: \(W_{net} = W_1 + W_2\) The net heat transfer is the sum of heat input in Step 1 and heat removed in Step 2: \(Q_{net} = Q_1 + Q_2\)

Calculate the net heat transfer

Using the given values, we can calculate the net heat transfer: \(Q_{net} = Q_1 + Q_2 = 45 - 60\) \(Q_{net} = -15 J\)

Calculate the work done in Step 2

Since the net work done is equal to the net heat transfer, we can write: \(-15 J = -10 J + W_2\) Now, we can solve for W2: \(W_2 = -15 J + 10 J\) \(W_2 = -5 J\) This means that the work done on the gas during the compression in Step 2 is -5 J (negative since it is work done on the gas).

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics that involves the movement of thermal energy from one place to another due to temperature differences. There are three main modes of heat transfer: conduction, which occurs through a material; convection, which occurs within fluids (liquids or gases) as they move; and radiation, which occurs via electromagnetic waves and does not require a medium.

In our exercise, heat transfer occurs during the two steps of the cyclic process. In Step 1, 45 J of heat is added to the gas, which implies an energy intake. Conversely, in Step 2, 60 J of heat is removed from the gas, indicating that the system has released energy. These two steps demonstrate the first law of thermodynamics, which states that energy can neither be created nor destroyed, only transferred or transformed.

Work Done by Gas

Work in thermodynamics is defined as the energy transfer that occurs when a force is applied over a distance. In the context of a gas, work is done when the gas expands or is compressed.

During the expansion in Step 1 of our exercise, work is done by the gas as it pushes outward against an external force, such as a piston in an engine. The expansion work is given as -10 J, with the negative sign indicating that the system is losing energy. This concept is vital as it showcases energy interaction in thermodynamic systems, where work can be either done by the gas on its surroundings (expansion work) or on the gas by the surroundings (compression work).

Cyclic Process

A cyclic process in thermodynamics is a series of transformations that return a system to its initial state. This means that at the end of the cycle, all the state properties, such as pressure, temperature, and volume, are unchanged from their initial values.

For a cyclic process, the first law of thermodynamics indicates that the net work done by the system over one complete cycle is equal to the net heat transfer into the system, as depicted in our exercise. The equation \( W_{net} = Q_{net} \) represents this relationship and is used to calculate the net changes in both heat and work over a cycle. In the given exercise, the gas undergoes a two-step cycle with specific heat transfers and work done in each step.

Gas Compression Work

Gas compression work refers to the work done on a gas when its volume decreases under pressure. This compression requires an external force, such as a piston compressing the gas, which transfers energy to the gas.

In Step 2 of our exercise, the gas is compressed, and we need to calculate the compression work. Compression work is analogous to expansion work but with a reversed sign convention; it is positive when we calculate the work done on the system. The calculated work for the gas compression in Step 2, which is -5 J, indicates energy transferred into the gas during the process.