A gas with initial state variables p₁, V₁, and T₁ is cooled in an isochoric process until p₂ = (1/3)p₁. What are V₂?

Verified step by step guidance

1. Step 1: Understand the problem. An isochoric process means the volume remains constant throughout the process. Therefore, V₂ = V₁. This is a key property of isochoric processes.
2. Step 2: Recall the ideal gas law, which is given by: $pV=nRT$. Since the volume (V) is constant, we can write the relationship between pressure and temperature as: $\frac{p}{T}=\backslash (\backslash text\{constant\}\backslash )$.
3. Step 3: Use the given condition that . Substitute this into the proportionality derived from the ideal gas law: $\frac{p_1}{T_1}=\frac{p_2}{T_2}$.
4. Step 4: Rearrange the equation to solve for $T_2$: $T_2=\frac{p_2}{T_1}{p}_1$. Substitute into the equation.
5. Step 5: Simplify the expression for $T_2$ to find its relationship with $T_1$. Since the volume remains constant, the final volume $V_2$ is equal to the initial volume $V_1$.

Key Concepts

Isochoric Process

An isochoric process is a thermodynamic process in which the volume of the system remains constant. During this process, any change in temperature or pressure occurs without any work being done on or by the system, as work is defined as force applied over a distance. This characteristic is crucial for understanding how pressure and temperature relate when volume does not change.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates pressure (p), volume (V), temperature (T), and the number of moles (n) of a gas. It is expressed as PV = nRT, where R is the ideal gas constant. This law allows us to predict how changes in one state variable affect the others, particularly useful in analyzing the behavior of gases during various processes.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its absolute temperature when pressure is held constant. In the context of an isochoric process, while the volume remains unchanged, understanding this law helps clarify how temperature and pressure interact. Since the volume does not change, a decrease in temperature will lead to a corresponding decrease in pressure, as indicated in the problem.