## Structure 1.5.4—The relationship between the pressure, volume, temperature and amount of an ideal gas is shown in the ideal gas equation PV = nRT and the combined gas law

Structure 1.5.4—The relationship between the pressure, volume, temperature and amount of an ideal gas is shown in the ideal gas equation PV = nRT and the combined gas law:

What You’ll Learn:

• Solve problems relating to the ideal gas equation.
• Units of volume and pressure should be SI only. The value of the gas constant R, the ideal gas equation, and the combined gas law, are given in the data booklet.

Keywords

ideal gas, moving particles, negligible volume, no intermolecular forces, elastic collisions, real gases, deviation, low temperature, high pressure, molar volume, constant, specific temperature, specific pressure, pressure, volume, temperature, amount, ideal gas equation, PV=nRT, combined gas law

Tool 1, Inquiry 2—How can the ideal gas law be used to calculate the molar mass of a gas from
experimental data?

The ideal gas equation, PV = nRT, and the combined gas law both describe the relationship between pressure (P), volume (V), temperature (T), and the amount of an ideal gas in moles (n). The main difference between the two is that the ideal gas equation includes the amount of gas (n) and the ideal gas constant (R), while the combined gas law does not.

1. Ideal Gas Equation (PV = nRT): The ideal gas equation represents the behavior of an ideal gas, a hypothetical gas that assumes negligible volume for individual gas particles and no intermolecular forces between them. This equation relates the pressure, volume, temperature, and the amount of gas in moles, where R is the ideal gas constant (8.314 J/(mol·K)). The ideal gas equation is applicable when dealing with changes in the amount of gas or when absolute values of the variables are required.
2. Combined Gas Law (P₁V₁/T₁ = P₂V₂/T₂): The combined gas law is derived from Boyle’s law (P₁V₁ = P₂V₂ at constant n and T), Charles’s law (V₁/T₁ = V₂/T₂ at constant n and P), and Gay-Lussac’s law (P₁/T₁ = P₂/T₂ at constant n and V). This law relates the initial state (P₁, V₁, T₁) to the final state (P₂, V₂, T₂) of a given gas sample undergoing changes in pressure, volume, and temperature, assuming that the amount of gas remains constant (n is constant). The combined gas law is useful when comparing two different sets of conditions for the same gas sample.

The combined gas law is derived by combining three fundamental gas laws: Boyle’s law, Charles’s law, and Gay-Lussac’s law. Each of these laws focuses on the relationship between two variables while holding the other two constant.

1. Boyle’s Law (P₁V₁ = P₂V₂ at constant n and T): Boyle’s law states that the pressure of a given amount of gas is inversely proportional to its volume when the temperature is held constant. Mathematically, this relationship is expressed as P₁V₁ = P₂V₂, where P₁ and V₁ are the initial pressure and volume, and P₂ and V₂ are the final pressure and volume.
2. Charles’s Law (V₁/T₁ = V₂/T₂ at constant n and P): Charles’s law states that the volume of a given amount of gas is directly proportional to its absolute temperature when the pressure is held constant. Mathematically, this relationship is expressed as V₁/T₁ = V₂/T₂, where V₁ and T₁ are the initial volume and temperature, and V₂ and T₂ are the final volume and temperature.
3. Gay-Lussac’s Law (P₁/T₁ = P₂/T₂ at constant n and V): Gay-Lussac’s law states that the pressure of a given amount of gas is directly proportional to its absolute temperature when the volume is held constant. Mathematically, this relationship is expressed as P₁/T₁ = P₂/T₂, where P₁ and T₁ are the initial pressure and temperature, and P₂ and T₂ are the final pressure and temperature.

To derive the combined gas law, we start with Boyle’s law and multiply both sides by T₁ and T₂:

P₁V₁T₂ = P₂V₂T₁

Now, we can rearrange the equation by dividing both sides by P₁V₁T₁:

(P₁V₁T₂) / (P₁V₁T₁) = (P₂V₂T₁) / (P₁V₁T₁)

This simplification results in the combined gas law:

P₁V₁/T₁ = P₂V₂/T₂

The combined gas law relates the initial state (P₁, V₁, T₁) to the final state (P₂, V₂, T₂) of a given gas sample undergoing changes in pressure, volume, and temperature, assuming that the amount of gas remains constant (n is constant).

Questions

1. A gas sample initially has a pressure of 100 kPa, a volume of 2 dm³, and a temperature of 25°C. If the pressure is increased to 200 kPa and the temperature to 50°C, what will be the new volume?
• P₁ = 100 kPa, V₁ = 2 dm³, T₁ = 25°C, P₂ = 200 kPa, T₂ = 50°C
• Calculate: V₂
2. A gas occupies a volume of 5 dm³ at 20°C and 150 kPa. If the volume is increased to 10 dm³ and the temperature to 40°C, what will be the new pressure?
• V₁ = 5 dm³, T₁ = 20°C, P₁ = 150 kPa, V₂ = 10 dm³, T₂ = 40°C
• Calculate: P₂
3. A gas has an initial pressure of 300 kPa, a volume of 8 dm³, and a temperature of 30°C. If the pressure is decreased to 150 kPa and the volume to 4 dm³, what will be the new temperature?
• P₁ = 300 kPa, V₁ = 8 dm³, T₁ = 30°C, P₂ = 150 kPa, V₂ = 4 dm³
• Calculate: T₂
4. A gas sample has an initial volume of 3 dm³ at 100 kPa and 25°C. If the pressure is increased to 200 kPa and the volume to 6 dm³, what will be the final temperature?
• V₁ = 3 dm³, P₁ = 100 kPa, T₁ = 25°C, V₂ = 6 dm³, P₂ = 200 kPa
• Calculate: T₂
5. A gas initially has a pressure of 120 kPa, a volume of 5 dm³, and a temperature of 50°C. If the pressure is decreased to 60 kPa and the temperature to 20°C, what will be the new volume?
• P₁ = 120 kPa, V₁ = 5 dm³, T₁ = 50°C, P₂ = 60 kPa, T₂ = 20°C
• Calculate: V₂
6. A gas sample occupies a volume of 10 dm³ at 30°C and 200 kPa. If the volume is decreased to 5 dm³ and the temperature to 10°C, what will be the new pressure?
• V₁ = 10 dm³, T₁ = 30°C, P₁ = 200 kPa, V₂ = 5 dm³, T₂ = 10°C
• Calculate: P₂
7. A gas has an initial pressure of 180 kPa, a volume of 7 dm³, and a temperature of 40°C. If the pressure is increased to 360 kPa and the temperature to 80°C, what will be the new volume?
• P₁ = 180 kPa, V₁ = 7 dm³, T₁ = 40°C, P₂ = 360 kPa, T₂ = 80°C
• Calculate: V₂
8. A gas sample initially has a volume of 12 dm³ at 50°C and 250 kPa. If the pressure is decreased to 125 kPa and the volume increased to 24 dm³, what will be the final temperature? V₁ = 12 dm³, T₁ = 50°C, P₁ = 250 kPa, V₂ = 24 dm³, P₂ = 125 kPa, Calculate: T₂
9. A gas initially has a pressure of 90 kPa, a volume of 4 dm³, and a temperature of 35°C. If the pressure is increased to 180 kPa while keeping the volume constant, what will be the new temperature?
• P₁ = 90 kPa, V₁ = 4 dm³, T₁ = 35°C, P₂ = 180 kPa, V₂ = 4 dm³
• Calculate: T₂
10. A gas sample has an initial pressure of 300 kPa, a volume of 6 dm³, and a temperature of 60°C. If the pressure is increased to 600 kPa and the temperature is decreased to 30°C, what will be the new volume? P₁ = 300 kPa, V₁ = 6 dm³, T₁ = 60°C, P₂ = 600 kPa, T₂ = 30°C, Calculate: V₂