Question

A \(1.00-\mathrm{L}\) volume of a gas undergoes first an isochoric process in which its pressure doubles, followed by an isothermal process until the original pressure is reached. Determine the final volume of the gas.

Short answer

Answer: The final volume of the gas is 0.50 L.

Step-by-step solution

Define the initial state of the gas

Let's call the initial pressure, volume, and temperature P1, V1, and T1, respectively. According to the problem, V1 is equal to 1.00 L.

Determine the pressure after the isochoric process

During the isochoric process, the volume remains constant (V1), and the pressure doubles. Let's denote the new pressure as P2, which can be calculated as: P2 = 2 * P1

Apply the ideal gas law to relate pressure, volume, and temperature in the isochoric process

Since the process is isochoric, we can apply the ideal gas law to relate P1, V1, T1, and P2. The number of moles (n) and gas constant (R) remain constant in both processes, so we write the following equation: P1 * V1 / T1 = P2 * V1 / T2 Since the volume remains constant, we can cancel V1 from both sides. P1 / T1 = P2 / T2

Determine the temperature after the isochoric process

Now, we need to find the value of T2 after the isochoric process. We know that P2 = 2 * P1. Substituting this into the equation above and solving for T2: T2 = 2 * P1 * T1 / P1 The P1 cancels out: T2 = 2 * T1

Apply the ideal gas law to relate pressure, volume, and temperature in the isothermal process

During the isothermal process, the temperature remains constant (T2), and the pressure returns to the original pressure (P1). Let's denote the final volume as V2. Using the ideal gas law: P1 * V1 / T1 = P1 * V2 / T2 Since P1 and T1 are the initial condition of the isothermal process, we can cancel P1 from both sides. V1 / T1 = V2 / T2 Now we can substitute T2, which we found in step 4: V1 / T1 = V2 / (2 * T1)

Determine the final volume of the gas, V2

Now we just need to find V2. We can cancel out T1 from both sides and multiply both sides by 2: V1 = 2 * V2 Finally, we can isolate V2: V2 = V1 / 2 Since V1 is equal to 1.00 L, the final volume V2 is: V2 = 1.00 L / 2 = 0.50 L The final volume of the gas after both processes is 0.50 L.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of a gas through the equation
\( PV = nRT \)
where P stands for pressure, V is the volume, n represents the number of moles, R is the universal gas constant, and T is the temperature in Kelvins. This equation describes how gases tend to behave under different sets of conditions, assuming the gas particles do not interact and take up no space, which is a good approximation for many gases under normal conditions.

In the context of the exercise, we use the ideal gas law to find relationships between the gas's pressure, volume, and temperature during both isochoric and isothermal processes. It allows us to determine changes in one variable while keeping others constant.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and the forms of energy involved in chemical and physical processes. At the core of thermodynamics are laws governing these interactions. The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another.

In an isochoric process, the volume of the gas remains constant. If heat is added to the system, it results in an increase in pressure and temperature but no work is done because volume does not change. Conversely, in an isothermal process, the temperature stays constant, which means that any heat transfer into or out of the system must result in work done, causing a change in volume while maintaining pressure-temperature equilibrium. These principles are directly applied when we solve problems related to gas behavior under different thermodynamic processes.

Gas Volume and Pressure Relationship

The relationship between gas volume and pressure is described by Boyle's Law, one of the gas laws stating that, for a given amount of gas at a constant temperature, the volume of the gas is inversely proportional to its pressure. This can be mathematically expressed as
\( P \times V = \text{constant} \)
This law is particularly relevant during isothermal processes where temperature is constant. In the given exercise, after the gas undergoes an isochoric process (constant volume), it enters an isothermal process. During the isothermal phase, as the original pressure is restored, the volume must change to compensate and maintain the product of pressure and volume. Understanding this inverse relationship allows students to predict the behavior of a gas when either its volume or pressure is altered while temperature remains stable.