Problem 32 - Algebra and Trigonometry

Question

Boyle's Law The pressure \(P\) of a sample of gas is directly proportional to the temperature \(T\) and inversely proportional to the volume \(V\) (a) Write an equation that expresses this variation. (b) Find the constant of proportionality if 100 \(\mathrm{L}\) of gas exerts a pressure of 33.2 \(\mathrm{kPa}\) at a temperature of 400 \(\mathrm{K}\) (absolute temperature measured on the Kelvin scale). (c) If the temperature is increased to 500 \(\mathrm{K}\) and the volume is decreased to 80 \(\mathrm{L}\) , what is the pressure of the gas?

Step-by-Step Solution

Answer

(a) \( P = k \cdot \frac{T}{V} \); (b) \( k = 8.3 \mathrm{kPa \cdot L/K} \); (c) Pressure is 51.875 \mathrm{kPa}.

Step 1: Understanding Boyle's Law

Boyle's Law states that the pressure of a gas is directly proportional to its temperature and inversely proportional to its volume. Mathematically, this can be expressed as \( P \propto \frac{T}{V} \), where \( P \) is the pressure, \( T \) is the temperature, and \( V \) is the volume.

Step 2: Formulating the Equation

To express the proportionality as an equation, introduce a constant of proportionality \( k \). Thus, the equation becomes \( P = k \cdot \frac{T}{V} \). This equation relates pressure, temperature, and volume with the constant \( k \).

Step 3: Determining the Constant of Proportionality

Given values are \( P = 33.2 \, \mathrm{kPa}, \ T = 400 \, \mathrm{K}, \) and \( V = 100 \, \mathrm{L} \). Plug these values into the equation \( P = k \cdot \frac{T}{V} \) to find \( k \): \[ 33.2 = k \cdot \frac{400}{100} \] Calculate \( k \):\[ k = \frac{33.2 \times 100}{400} = 8.3 \mathrm{kPa \cdot L/K} \] .

Step 4: Calculating Pressure with New Conditions

Using the equation \( P = k \cdot \frac{T}{V} \) with the new values \( T = 500 \, \mathrm{K} \) and \( V = 80 \, \mathrm{L} \), and the previously found \( k = 8.3 \, \mathrm{kPa \cdot L/K} \), calculate the new pressure: \[ P = 8.3 \cdot \frac{500}{80} \] Calculate \( P \):\[ P = 8.3 \cdot 6.25 = 51.875 \mathrm{kPa} \].

Key Concepts

Gas Laws

Gas laws help us understand how gases behave under various conditions. One of the fundamental gas laws is Boyle's Law. This law explores the relationship between pressure, temperature, and volume for a gas. More specifically, it shows how changes in one of these variables affect the others.
Gas laws are essential in science because they allow us to calculate how a gas will respond to changes in its environment. These laws provide rules for predicting and explaining the behavior of gases under different conditions:

• Pressure (\(P\)) refers to the force that the gas exerts on the walls of its container.
• Temperature (\(T\)) is a measure of the average kinetic energy of the gas molecules.
• Volume (\(V\)) is the space the gas occupies.

Understanding these relationships is crucial for fields like chemistry and physics, where predicting gas behavior is necessary for many applications.

Proportionality Constant

At the heart of Boyle's Law, the concept of a proportionality constant (\(k\)) is very practical. When we say that pressure is directly proportional to the temperature and inversely proportional to the volume, we are describing a relationship that can be summarized by an equation.
To express this proportional relationship in a calculable way, we introduce \(k\). It acts as a numerical factor linking pressure, temperature, and volume. The equation is expressed as:\[ P = k \cdot \frac{T}{V} \]Let's recap:

• \(k\) remains the same as long as conditions like the type of gas and mass of the gas stay constant.
• This constant helps predict how pressure will change if either temperature or volume changes.

Knowing \(k\) is like having a key that deciphers the relationship between these three important gas variables.

Pressure Calculation

Calculating pressure is an essential application of Boyle's Law. When the conditions of a gas test change, like temperature or volume, we use the law to find the new pressure.
Here's how we do it:
Given the initial conditions, we can calculate the proportionality constant \(k\) using the measured values of pressure, temperature, and volume. Once \(k\) has been found, foresee alterations in the gas conditions by transposing these into the equation. For instance:
If the temperature increases and volume reduces, according to:\[ P = k \cdot \frac{T}{V} \] Use new values to compute the updated pressure. This operation provides insight into how gases will respond in practical scenarios, such as adjusting conditions in a laboratory or industrial environment.

Temperature and Volume Relationship

The relationship between temperature and volume is another core element of understanding gas behavior. According to Boyle's Law, when temperature changes, volume changes too, impacting pressure.
This can feel complex but remember these points:

• As temperature (\(T\)) increases with volume constant, the pressure typically increases.
• If the volume (\(V\)) decreases with temperature constant, pressure tends to increase.
• When both temperature and volume change together, their combined effect on pressure can be calculated using the formula \[ P = k \cdot \frac{T}{V} \].

This relationship shows the intricate balance between these two variables. Understanding this allows you to predict what may occur with the pressure based on alterations to temperature and volume. This is critical for both experimental and theoretical modeling in the sciences.