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F3.4 · Describe relationships among ideal-gas variables

Learn to describe relationships among ideal-gas variables through clear examples and targeted practice.

Ontario Grade 11 Chemistry

Gases and Atmospheric Chemistry

How pressure, volume, temperature, and amount change together

A sealed syringe becomes harder to push as its trapped gas is squeezed into less space. A flexible container can expand when its gas is warmed. These familiar changes show that gas properties are connected. The ideal-gas model is a simplified way to describe these connections. It relates pressure, volume, temperature, and amount of gas. To describe a relationship clearly, state which other variables are held constant.

What you will learn

Identify the four variables used to describe an ideal gas.
Describe how two gas variables relate when the relevant other variables are held constant.
Use kelvin temperatures and consistent units in gas relationships.
Choose and use an equation that matches the stated conditions.

1. From observations to gas variables

A variable is a quantity that can be measured or changed. Four variables describe the state of an ideal gas: pressure, volume, temperature, and amount. Pressure describes how strongly a gas pushes on a surface. Volume is the space the gas occupies. Temperature tells how hot or cold the gas is. Amount tells how much gas is present and is commonly measured in moles.

The symbols are

P

for pressure,

V

for volume,

T

for absolute temperature, and

n

for amount in moles. Absolute temperature is measured from a scale that starts at the lowest possible temperature. In gas relationships, it is measured in kelvins, with the unit symbol K.

At the particle level, a gas consists of particles moving through the available space. Collisions of particles with the container walls help explain pressure. Squeezing a sealed syringe leaves the same amount of gas in less space. The particles have less room, and pressure rises. This observation fits the ideal-gas model.

A relationship describes how one variable changes when another changes under stated conditions. A direct relationship means that the variables change in the same direction. An inverse relationship means that one increases while the other decreases. These terms are meaningful only when the relevant other variables are held constant.

The four variables are pressure, volume, temperature, and amount.
Describe a relationship together with the conditions that stay constant.
Particle collisions with container walls help explain gas pressure.

2. How pairs of variables relate

For a fixed amount of gas at constant temperature, pressure and volume are inversely related. If the gas occupies less volume, its pressure is greater. If it occupies more volume, its pressure is lower. This relationship is commonly called Boyle’s law. The product of pressure and volume remains constant for the same amount of gas at the same temperature.

For a fixed amount of gas at constant pressure, volume and absolute temperature are directly related. If the temperature in kelvins increases, the volume increases. If the temperature decreases, the volume decreases. This relationship is commonly called Charles’s law.

For a fixed amount of gas at constant volume, pressure and absolute temperature are directly related. A higher temperature in kelvins corresponds to a higher pressure. A lower temperature corresponds to a lower pressure. This relationship is commonly called Gay-Lussac’s law. A rigid, sealed container is an example of a situation where volume and amount can stay fixed.

At constant pressure and temperature, volume and amount are directly related. A greater amount of gas occupies a greater volume under those same conditions. This relationship is commonly called Avogadro’s law.

For gas calculations, use kelvins rather than Celsius temperatures. To convert a Celsius temperature, add 273.15. For example, a Celsius temperature of 20.0 °C corresponds to 293.15 K. A change of one degree Celsius has the same size as a change of one kelvin, but the scales have different zero points.

PV=k,\quad \frac{V}{T}=k,\quad \frac{P}{T}=k,\quad \frac{V}{n}=k

At constant temperature and amount, pressure and volume are inversely related.
At constant pressure and amount, volume and kelvin temperature are directly related.
At constant volume and amount, pressure and kelvin temperature are directly related.
At constant pressure and temperature, volume and amount are directly related.

3. Equations for comparing gas conditions

The symbol

k

stands for a constant value in a particular relationship. Its value can differ from one relationship or gas sample to another. These equations summarize how pairs of variables relate when the stated conditions remain fixed.

When the amount of the same gas sample stays fixed but pressure, volume, and temperature change, the combined gas law compares the starting state with the later state. State 1 represents the starting conditions; state 2 represents the later conditions. Use the same pressure units in both states and the same volume units in both states.

The ideal-gas equation brings all four variables together for one state of an ideal gas. In it,

R

is the ideal gas constant. Its numerical value has units, so the chosen value of

R

must match the pressure and volume units in the calculation. Do not use this equation with incompatible units.

Before calculating, list the known values and identify what stays fixed. Choose the relationship that matches those conditions. Convert temperatures to kelvins, rearrange with algebra if needed, substitute values with units, and check that the answer has the requested unit. The direction of change is also a useful check: compressing a fixed amount of gas at constant temperature should raise its pressure.

\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2},\qquad PV=nRT

Use the combined gas law to compare two states when the amount of gas is fixed.
Use the ideal-gas equation to relate all four variables in one state.
Use kelvins and units that are compatible throughout a calculation.

4. Describe and check relationships

A complete description names the changing variables and the fixed conditions. For example, say that pressure and volume are inversely related for a fixed amount of gas at constant temperature. Saying only that pressure and volume are related leaves out the condition that makes the statement true.

When more than one variable changes, do not apply a pair relationship without checking its conditions. For example, if both volume and temperature change while the amount stays fixed, use the combined gas law to compare the two states. This avoids treating a relationship as though a variable were constant when it is not.

Check the result in two ways. First, check the units: pressure needs a pressure unit, volume needs a volume unit, and temperature values in gas equations must be in kelvins. Second, check whether the direction makes sense given the stated conditions. A calculation can be set up correctly only if it matches those conditions.

Include the fixed conditions whenever you describe a gas relationship.
Do not use a pair relationship if its required conditions are changing.
Check units and the expected direction of change.

Worked example

Finding the new pressure

A fixed amount of gas occupies 2.40 L at 98.0 kPa and 300. K. It is moved to conditions where its volume is 1.80 L and its temperature is 330. K. Find its new pressure.

Choose the relationship
The amount of gas stays fixed, while pressure, volume, and temperature can change. Use the combined gas law. The temperatures are already in kelvins, and the pressure and volume units are consistent between the two states.
$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$
Rearrange for pressure
Make the final pressure the subject. This places the known starting values and the final volume and temperature in the expression used to calculate it.
$P_2=\frac{P_1V_1T_2}{T_1V_2}$
Substitute and calculate
Substitute each value with its unit. The litre and kelvin units cancel in the ratios, leaving kilopascals. The given measurements have three significant figures, so report the result to three significant figures.
$P_2=\frac{(98.0\ \mathrm{kPa})(2.40\ \mathrm{L})(330.\ \mathrm{K})}{(300.\ \mathrm{K})(1.80\ \mathrm{L})}=143.733\ldots\ \mathrm{kPa}\approx144\ \mathrm{kPa}$

Answer: The new pressure is 144 kPa.

Check: The volume decreases and the temperature increases. For the same amount of gas, both changes support a higher pressure. The calculated pressure is higher than the starting pressure, so its direction is reasonable.

Common mistakes and how to avoid them

Using a Celsius temperature directly in a gas equation.

Correction: Convert Celsius to kelvins before using a gas relationship.

Calling pressure and volume directly related.

Correction: For a fixed amount of gas at constant temperature, pressure and volume are inversely related.

Describing a relationship without stating what stays constant.

Correction: Name the fixed conditions, such as constant temperature and amount.

Reporting a number without a unit or with too many significant figures.

Correction: Include the appropriate unit and round to the precision supported by the given measurements.

Lesson summary

An ideal gas is described by pressure, volume, temperature, and amount.
Pressure and volume are inversely related at constant temperature and amount.
Volume and kelvin temperature are directly related at constant pressure and amount.
Pressure and kelvin temperature are directly related at constant volume and amount.
Volume and amount are directly related at constant pressure and temperature.
Use kelvins, consistent units, and the equation that matches the stated conditions.

Check your understanding

Question 1

A fixed amount of gas is warmed in a rigid, sealed container. What happens to its pressure?

It increases.
It decreases.
It stays the same.
correctIndex

Show answer and explanation

It increases.

At constant volume and amount, pressure is directly related to kelvin temperature. Warming the gas increases its pressure.

Question 2

Which temperature scale should be used in gas relationships?

Celsius temperature without conversion
Kelvin temperature
Either scale without conversion
correctIndex

Show answer and explanation

Kelvin temperature

Gas relationships use absolute temperature, measured in kelvins. Celsius temperatures must be converted first.

Question 3

At constant pressure and temperature, a gas sample contains more moles. What happens to its volume?

It becomes smaller.
It becomes larger.
It does not change.
correctIndex

Show answer and explanation

It becomes larger.

At constant pressure and temperature, volume and amount are directly related. More gas occupies a larger volume.

Key terms

Ideal-gas model: A simplified model that describes relationships among the pressure, volume, temperature, and amount of a gas.
Pressure: A measure of how strongly a gas pushes on a surface.
Volume: The space occupied by a gas.
Absolute temperature: Temperature measured in kelvins for use in gas relationships.
Amount: The quantity of gas, commonly measured in moles.
Direct relationship: A relationship in which two variables change in the same direction when the relevant other variables are fixed.
Inverse relationship: A relationship in which one variable increases as the other decreases when the relevant other variables are fixed.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Chemistry (SCH3U), expectation F3.4. It is a study resource, not an official curriculum publication.

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