Boyle's Law and Charles's Law Revision Notes for HSC SSCE Chemistry

Boyle's Law and Charles's Law

Introduction

When studying gases, we need to understand how they behave when conditions like temperature and pressure change. This is where the gas laws become essential. In this note, we'll explore two fundamental gas laws: Boyle's Law and Charles's Law. These laws help us predict and calculate how gas volumes respond to changes in pressure and temperature.

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Understanding gas laws is crucial for many real-world applications, from predicting weather patterns to designing engines and understanding respiratory physiology. These fundamental principles form the foundation of thermodynamics and physical chemistry.

Understanding Boyle's law

What is Boyle's law?

Boyle's Law describes the relationship between the pressure and volume of a gas when temperature remains constant. Robert Boyle discovered in $1662$ that when you compress a gas (increase its pressure), its volume decreases proportionally, and vice versa.

The key principle is this: for a fixed amount of gas at constant temperature, the product of pressure and volume always equals the same constant value.

Mathematical expression

Boyle's Law can be written as:

$PV = k$

where:

$P$ = pressure of the gas
$V$ = volume of the gas
$k$ = a constant value

This means that pressure and volume are inversely proportional. If you double the pressure, the volume halves. If you reduce the pressure to one-fifth of its original value, the volume increases five times.

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The inverse relationship means that as one variable increases, the other decreases by the same factor. This is different from a direct relationship where both variables change in the same direction.

Practical form of Boyle's law

A more useful form of this equation for calculations is:

$P_2V_2 = P_1V_1$

where:

$P_1$ and $V_1$ = initial pressure and volume
$P_2$ and $V_2$ = final pressure and volume

Both products equal the same constant $k$ . You can use any units for pressure and volume, as long as you use the same units on both sides of the equation.

Investigating Boyle's law

The relationship between pressure and volume can be investigated experimentally using a simple apparatus. A sealed syringe containing air is clamped vertically, and weights (such as paving bricks or heavy books) are placed on top. The additional weight increases the pressure on the gas inside.

As you add more weight, the gas volume decreases. By recording different weights and corresponding volumes, you can verify that $PV$ remains approximately constant.

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In practice, the product $PV$ might not be perfectly constant due to experimental limitations such as temperature fluctuations, gas leaks, or measurement errors. However, the trend will clearly demonstrate the inverse relationship.

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Worked Example: Boyle's Law Calculation

Problem: A sample of gas originally had a volume of $1.50$ L at $20°\text{C}$ and $100$ kPa pressure. What volume would it have at $400$ kPa pressure?

Solution:

Using $P_2V_2 = P_1V_1$ :

Given:

$P_1 = 100$ kPa
$V_1 = 1.50$ L
$P_2 = 400$ kPa
$V_2 = ?$

Substituting values: $(400 \text{ kPa}) \times V_2 = (100 \text{ kPa}) \times 1.50 \text{ L}$

$V_2 = \frac{100 \times 1.50 \text{ kPa L}}{400 \text{ kPa}}$

$V_2 = 0.38 \text{ L}$

Answer: The volume decreases from $1.50$ L to $0.38$ L when pressure increases from $100$ kPa to $400$ kPa, demonstrating the inverse relationship.

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Exam Tip for Boyle's Law

Always check your answer makes physical sense. If pressure increases, volume should decrease (and vice versa). Include units in your calculations to ensure they cancel correctly.

Understanding Charles's law

What is Charles's law?

Charles's Law describes how the volume of a gas changes with temperature when pressure remains constant. French physicist Jacques Charles discovered in $1787$ that gas volume increases linearly with temperature.

The absolute temperature scale

Before we can properly express Charles's Law, we need to understand the absolute temperature scale (also called the Kelvin scale). When scientists plotted volume versus temperature for different gases and extrapolated their data, they found that all lines intersected the temperature axis at $-273°\text{C}$ (more precisely $-273.15°\text{C}$ ).

This led to the development of the Kelvin scale, where:

$0$ K = $-273.15°\text{C}$ (absolute zero)
$273.15$ K = $0°\text{C}$
$373.15$ K = $100°\text{C}$

The conversion formula is:

$T(\text{K}) = T(°\text{C}) + 273.15$

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One major advantage of the Kelvin scale is that all physically accessible temperatures have positive values. You cannot have a temperature below $0$ K - it's the absolute zero of temperature, the point at which all molecular motion theoretically ceases.

Mathematical expression

When we replot volume versus temperature using the Kelvin scale instead of Celsius, all the lines pass through the origin. This gives us Charles's Law:

$V = kT$

where:

$V$ = volume of the gas
$T$ = absolute temperature (in kelvin)
$k$ = a constant value

This tells us that at constant pressure, the volume of a fixed quantity of gas is directly proportional to its absolute temperature.

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The direct proportionality means that if you double the absolute temperature, you double the volume. This linear relationship only works when using the Kelvin scale - it would not work with Celsius or Fahrenheit!

Practical form of Charles's law

A more useful form for calculations is:

$\frac{V_2}{T_2} = \frac{V_1}{T_1}$

where:

$V_1$ and $T_1$ = initial volume and temperature
$V_2$ and $T_2$ = final volume and temperature

Each quotient equals the same constant $k$ . You can use any units for volume (as long as they're consistent), but temperature must always be in kelvin.

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Critical Reminder: Temperature MUST be in kelvin for Charles's Law calculations. Using Celsius will give completely incorrect results because Charles's Law depends on the direct proportionality that only exists when using absolute temperature.

Investigating Charles's law

You can investigate this relationship by sealing air in a syringe with a thermometer attached, then exposing it to different temperatures (ice bath, room temperature, hot water at various temperatures, boiling water). As temperature increases, the gas volume increases proportionally.

Experimental condition	Temperature (°C)	Volume (mL)
Ice bath
Room temperature
Hot water A
Hot water B
Boiling water

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Worked Example: Charles's Law Calculation

Problem: A sample of gas at $101$ kPa pressure had a volume of $2.5$ L at $100°\text{C}$ . What would its volume be at $0°\text{C}$ at the same pressure?

Solution:

Using $\frac{V_2}{T_2} = \frac{V_1}{T_1}$ :

First, convert temperatures to kelvin:

$T_1 = 100 + 273.2 = 373.2$ K
$T_2 = 0 + 273.2 = 273.2$ K
$V_1 = 2.5$ L

Substituting values: $\frac{V_2}{273.2 \text{ K}} = \frac{2.5 \text{ L}}{373.2 \text{ K}}$

$V_2 = \frac{2.5 \times 273.2 \text{ L K}}{373.2 \text{ K}}$

$V_2 = 1.8 \text{ L}$

Answer: The volume decreases from $2.5$ L to $1.8$ L when temperature decreases from $373.2$ K to $273.2$ K.

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Exam Tip for Charles's Law

Always convert Celsius temperatures to kelvin before using Charles's Law equations. Forgetting this step is the most common mistake that will give you incorrect answers. Make this your first step in every Charles's Law problem!

The combined gas law

Combining Boyle's and Charles's laws

When both pressure and temperature change simultaneously, we need to combine both laws. This gives us the combined gas law:

$\frac{PV}{T} = k$

where $k$ is a constant for a fixed quantity of gas.

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The combined gas law is essentially a merger of Boyle's Law and Charles's Law. It accounts for situations where we can't hold either temperature or pressure constant, which is common in real-world applications.

Practical form

The most useful form for calculations is:

$\frac{P_2V_2}{T_2} = \frac{P_1V_1}{T_1}$

This equation works because both sides equal the same constant $k$ . Pressure and volume can use any units (as long as they're consistent), but temperature must be in kelvin.

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Worked Example: Combined Gas Law Calculation

Problem: A certain quantity of gas had a volume of $1.30$ L at $101$ kPa and $80°\text{C}$ . What pressure is needed to compress it to $500$ mL at $30°\text{C}$ ?

Solution:

Using $\frac{P_2V_2}{T_2} = \frac{P_1V_1}{T_1}$ :

Given:

$P_1 = 101$ kPa
$V_1 = 1.3$ L
$V_2 = 500$ mL = $0.5$ L (convert to same units)
$T_1 = 80 + 273.2 = 353.2$ K
$T_2 = 30 + 273.2 = 303.2$ K

Substituting values: $\frac{P_2 \times 0.50 \text{ L}}{303.2 \text{ K}} = \frac{101 \text{ kPa} \times 1.3 \text{ L}}{353.2 \text{ K}}$

$P_2 = \frac{101 \times 1.3 \times 303.2 \text{ kPa L K}}{353.2 \times 0.50 \text{ K L}}$

$P_2 = 225 \text{ kPa}$

Answer: The pressure must increase to $225$ kPa to achieve the desired compression at the lower temperature.

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Exam Tip for Combined Gas Law

Substitute units along with numbers into the equation. This helps you verify that units cancel correctly to give the answer in the correct units. Also, always check that your answer makes physical sense based on the changes in conditions.

Practice problems

Here are some problems to test your understanding:

Boyle's law problems

	$P_1$	$V_1$	$P_2$	$V_2$
a	$344$ kPa	$505$ mL	$83.1$ kPa	?
b	$87$ kPa	$4.6$ L	?	$6.5$ L
c	$0.42$ kPa	$327$ mL	$846$ Pa	?
d	$231$ Pa	$4.55$ L	?	$653$ mL

Charles's law problems

	$V_1$	TEMP $_1$	$V_2$	TEMP $_2$
a	$500$ mL	$200$ K	?	$500$ K
b	$1.5$ L	$350$ K	$450$ mL	?
c	$250$ mL	$22°\text{C}$	?	$85°\text{C}$
d	$550$ mL	$180°\text{C}$	$1.35$ L	?

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When working through these problems, remember to:

Check if temperatures need conversion to kelvin
Ensure all units are consistent on both sides of the equation
Verify your answer makes physical sense given the changes in conditions

Connection to Avogadro's law

While not the main focus of this note, it's worth mentioning that Avogadro's Law relates volume to the amount of gas (in moles). At constant temperature and pressure:

$V = kn$

where $n$ is the number of moles of gas. This means that if you double the amount of gas, you double the volume (at constant temperature and pressure).

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Together, Boyle's Law, Charles's Law, and Avogadro's Law can be combined to form the Ideal Gas Law: $PV = nRT$ , where $R$ is the universal gas constant. This powerful equation relates all the gas variables in one expression.

Key experimental apparatus

When testing Boyle's Law, scientists use apparatus such as a syringe connected to a pressure gauge. This allows simultaneous measurement of pressure and volume as the syringe plunger is adjusted.

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Modern digital sensors can provide more accurate measurements than traditional apparatus, allowing for better verification of the gas laws. Computer data-logging systems can record hundreds of data points to show the precise relationship between variables.

Remember!

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Key Points to Remember:

Boyle's Law ( $P_2V_2 = P_1V_1$ ): For a fixed amount of gas at constant temperature, pressure and volume are inversely proportional. Doubling pressure halves volume.
Charles's Law ( $\frac{V_2}{T_2} = \frac{V_1}{T_1}$ ): For a fixed amount of gas at constant pressure, volume is directly proportional to absolute temperature. Always use Kelvin for temperature calculations.
Temperature conversion: $T(\text{K}) = T(°\text{C}) + 273.15$ . Absolute zero is $0$ K = $-273.15°\text{C}$ .
Combined Gas Law ( $\frac{P_2V_2}{T_2} = \frac{P_1V_1}{T_1}$ ): Use this when both pressure and temperature change. Temperature must be in kelvin.
Units: Pressure and volume can use any consistent units, but temperature must always be in kelvin for gas law calculations.
Check your work: Always verify that your answer makes physical sense based on the relationships described by each law.

Boyle's Law and Charles's Law (HSC SSCE Chemistry): Revision Notes