Interactive Gas Laws Laboratory
Combined Gas Law
The **Combined Gas Law** merges Boyle\'s Law, Charles\'s Law, and Gay-Lussac\'s Law. It defines the relationship between pressure, volume, and absolute temperature for a closed system containing a constant amount of gas.
Combined Gas Law Simulator
1. The Combined Gas Law Concept
The **Combined Gas Law** governs the behavior of a fixed amount of gas (constant moles \(n\)) when its pressure (\(P\)), volume (\(V\)), and absolute temperature (\(T\)) all change simultaneously.
Rather than restricting the gas to constant temperature (Boyle\'s Law), constant pressure (Charles\'s Law), or constant volume (Gay-Lussac\'s Law), the Combined Gas Law allows all three variables to vary together. This is a much more realistic model of everyday environments—such as a weather balloon expanding as it ascends into the cold upper atmosphere, or a car tire warming up and expanding slightly while driving on hot highways.
2. Mathematical Expression
The Combined Gas Law states that the product of pressure and volume divided by absolute temperature is constant for a closed gas sample:
For a gas sample changing from an initial state (subscript 1) to a final state (subscript 2), the law is expressed as:
Where:
- P₁, P₂ are the initial and final absolute pressures (atm, kPa, bar, etc.).
- V₁, V₂ are the initial and final volumes (L, mL, m³, etc.).
- T₁, T₂ are the initial and final absolute temperatures in Kelvin.
Rule of thumb: Pressure and volume units must match on both sides of the equation. Temperature must always be converted to Kelvin: \(T(K) = T(°C) + 273.15\).
3. Microscopic Origin (Kinetic Theory)
From the molecular perspective of **Kinetic Molecular Theory**:
1. Pressure is caused by molecules colliding with container boundaries.
2. Absolute temperature represents the average molecular kinetic energy. Higher temperature means faster molecules.
3. If we heat a gas (increasing T), molecules speed up.
4. If the volume (V) expands at the same time, molecules have more space to travel, which decreases collision frequency.
5. The final pressure (P) depends on the balance between these two competing effects: the increased velocity of the particles pushing outward, and the larger surface area spreading collisions thin.
4. Solved Mathematical Problems
- State the Combined Gas Law: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).
- Identify the given values: - Initial state: \(P_1 = 1.00 \text{ atm}\), \(V_1 = 4.00 \text{ L}\), \(T_1 = 300 \text{ K}\) - Final state: \(V_2 = 2.00 \text{ L}\), \(T_2 = 450 \text{ K}\)
- Rearrange the equation to solve for the final pressure \(P_2\): \(P_2 = \frac{P_1 \times V_1 \times T_2}{T_1 \times V_2}\).
- Substitute the values into the equation: \(P_2 = \frac{1.00 \times 4.00 \times 450}{300 \times 2.00}\).
- Calculate the result: - Numerator: \(1.00 \times 4.00 \times 450 = 1800\) - Denominator: \(300 \times 2.00 = 600\) - \(P_2 = \frac{1800}{600} = 3.00 \text{ atm}\).
- Conclude: The final pressure inside the cylinder is 3.00 atm.
- Recall the Combined Gas Law: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).
- Convert Celsius temperatures to Kelvin: - Initial temperature: \(T_1 = 20.0 + 273.15 = 293.15 \text{ K}\) - Final temperature: \(T_2 = -55.0 + 273.15 = 218.15 \text{ K}\)
- Identify other given values: - \(P_1 = 1.00 \text{ atm}\), \(V_1 = 2.00 \text{ L}\), \(P_2 = 0.25 \text{ atm}\)
- Rearrange the formula to solve for the final volume \(V_2\): \(V_2 = \frac{P_1 \times V_1 \times T_2}{T_1 \times P_2}\).
- Substitute the values: \(V_2 = \frac{1.00 \times 2.00 \times 218.15}{293.15 \times 0.25}\).
- Calculate the result: - Numerator: \(436.3\) - Denominator: \(73.2875\) - \(V_2 = \frac{436.3}{73.2875} \approx 5.95 \text{ L}\).
- Conclude: The weather balloon expands to a final volume of approximately 5.95 L.
- Use the Combined Gas Law: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).
- Convert temperatures to Kelvin: - \(T_1 = 10.0 + 273.15 = 283.15 \text{ K}\) - \(T_2 = 50.0 + 273.15 = 323.15 \text{ K}\)
- Identify other given values: - \(P_1 = 2.20 \text{ bar}\), \(V_1 = 20.0 \text{ L}\), \(V_2 = 20.2 \text{ L}\)
- Rearrange the formula to solve for the final pressure \(P_2\): \(P_2 = \frac{P_1 \times V_1 \times T_2}{T_1 \times V_2}\).
- Substitute the values: \(P_2 = \frac{2.20 \times 20.0 \times 323.15}{283.15 \times 20.2}\).
- Calculate the result: - Numerator: \(14218.6\) - Denominator: \(5719.63\) - \(P_2 = \frac{14218.6}{5719.63} \approx 2.486 \text{ bar}\).
- Conclude: The final pressure increases to approximately 2.49 bar.
5. Practice Questions
6. Frequently Asked Questions (FAQs)
What is the Combined Gas Law?
The Combined Gas Law is a gas law that merges Boyle's Law, Charles's Law, and Gay-Lussac's Law, stating that the ratio of the product of pressure and volume to absolute temperature is constant for a closed gas sample.
What is the formula for the Combined Gas Law?
The formula is P₁V₁/T₁ = P₂V₂/T₂, where P is pressure, V is volume, T is absolute temperature (Kelvin), and subscripts 1 and 2 refer to the initial and final states.
What variable is held constant in the Combined Gas Law?
The amount of gas (moles or mass) is held constant. The system must be closed, meaning no gas enters or escapes.
Can I use Celsius in the Combined Gas Law?
No. Just like all other gas laws, you must convert all Celsius temperatures to Kelvin by adding 273.15, as the law depends on the absolute thermodynamic scale.
How does Charles's Law relate to the Combined Gas Law?
Charles's Law occurs at constant pressure (P₁ = P₂). If we cancel pressure from the Combined Gas Law, we get Charles's Law: V₁/T₁ = V₂/T₂.
What units can be used for pressure in this law?
Any units for pressure are acceptable (atm, kPa, bar, mmHg, torr, or Pa), provided they are identical on both sides of the equation.
What happens to the volume of a gas if both its pressure and absolute temperature are doubled?
If pressure is doubled (2×) and absolute temperature is doubled (2×), the volume remains unchanged: V₂ = V₁ × (P₁/P₂) × (T₂/T₁) = V₁ × (1/2) × (2/1) = V₁.
Does the Combined Gas Law apply to ideal gases only?
Yes, it assumes ideal gas behavior. Under ordinary conditions, real gases follow it closely, but they deviate at extremely high pressures or near liquefaction temperatures.
How does Dalton's Law relate to this?
If a gas mixture is used, the Combined Gas Law still applies to the total pressure of the gas mixture, provided the total moles of gas remain constant.
What is the graphical representation of the Combined Gas Law?
Because it involves three variables (P, V, T), it is represented graphically in three dimensions. Projections onto 2D axes show hyperbolas (P-V isothermal lines) or straight lines (P-T and V-T lines).