Problem 19 - Calculus

Question

When a gas expands without gain or loss of heat, the rate of change of pressure with respect to volume is proportional to pressure divided by volume. Find a law connecting pressure and volume in this case.

Step-by-Step Solution

Answer

The law connecting pressure and volume is \( PV^k = C \).

Step 1: Understanding the Problem

We need to find a relationship between pressure \( P \) and volume \( V \) when a gas expands adiabatically, meaning without gain or loss of heat. It is given that the rate of change of pressure with respect to volume, \( \frac{dP}{dV} \), is proportional to \( \frac{P}{V} \). This can be expressed mathematically as \( \frac{dP}{dV} = -k \frac{P}{V} \), where \( k \) is a proportionality constant.

Step 2: Setup the Differential Equation

We have the differential equation \( \frac{dP}{dV} = -k \frac{P}{V} \). This is a separable differential equation, meaning we can separate the variables on either side of the equation. To do this, multiply both sides by \( V \) and divide both sides by \( P \): \( \frac{dP}{P} = -k \frac{dV}{V} \).

Step 3: Integrate Both Sides

Integrate both sides of the equation: \( \int \frac{dP}{P} = \int -k \frac{dV}{V} \). This gives us \( \ln |P| = -k \ln |V| + C \), where \( C \) is the integration constant. The integration results in natural logarithms because the integral of \( \frac{1}{X} \) is \( \ln |X| \).

Step 4: Exponentiate to Remove Logarithms

To remove the logarithms, exponentiate both sides of the equation: \( e^{\ln |P|} = e^{-k \ln |V| + C} \). This simplifies to \( |P| = e^C |V|^{-k} \). Since \( e^C \) is just another constant, we can call it \( C' \). Thus, \( P = C' V^{-k} \).

Step 5: Simplify and Write the Final Law

The final relationship can be simplified to \( P V^k = C' \), where \( C' \) is a constant. This is the adiabatic law relating pressure and volume for the given condition, often expressed as \( PV^\gamma = C \) in thermodynamics, where \( \gamma = k \).

Key Concepts

Differential Equation

A differential equation is a mathematical equation that involves functions and their derivatives. In the context of an adiabatic process, we are looking at how the pressure of a gas changes as its volume changes. Specifically, we know that the rate of change of pressure with respect to volume is proportional to pressure divided by volume. This can be written as:

• \( \frac{dP}{dV} = -k \frac{P}{V} \)

where \( \frac{dP}{dV} \) is the derivative of pressure with respect to volume, and \( k \) is a proportionality constant. The negative sign indicates that as volume increases, pressure decreases.

To solve this differential equation, we separate the variables so that all terms involving \( P \) are on one side and all terms involving \( V \) are on the other:

• \( \frac{dP}{P} = -k \frac{dV}{V} \)

This is a classic technique for solving separable differential equations, leading us to integrate both sides to find the relationship between \( P \) and \( V \).

Pressure-Volume Relationship

In an adiabatic process, there is a specific relationship between pressure and volume. This relationship is key in understanding how the system behaves without exchanging heat with its surroundings. Simplifying this relationship leads us to the law often expressed as \( PV^\gamma = C \), where \( \gamma = k \).

Applying integration to the separated differential equation \( \frac{dP}{P} = -k \frac{dV}{V} \), we get:

• \( \ln |P| = -k \ln |V| + C \)

By exponentiating, or getting rid of the natural logarithm through raising \( e \) to the power of each side, we simplify the equation to:

• \( P = C' V^{-k} \)

where \( C' \) is a new constant representing \( e^C \). Translating this into the more familiar form used in physics, we use \( \gamma \) instead of \( k \) to denote specific heat ratios in thermodynamics.

Thermodynamics Law

Thermodynamics laws describe how physical systems behave in terms of energy transfers and states. Here, we focus on the adiabatic process, a key phenomenon in thermodynamics. In this particular process, there's no heat exchange with the environment, which leads to an intrinsic pressure-volume relationship.

The law derived in this context is known as the adiabatic law \( PV^\gamma = C \). This equation stands for situations where a gas expands or compresses without gaining or losing heat. The constant \( \gamma \) is significant as it represents the ratio of specific heats (heat needed to raise temperature) at constant pressure to that at constant volume.

• It characterizes how easily a gas can compress or expand.
• This relationship is vital in technologies like engines and air compression systems.

The adiabatic law helps predict how changes in pressure and volume will jointly alter without external thermal influence, thus being a cornerstone in the field of thermodynamics.