The Ideal Gas Law Derivations: Videos & Practice Problems

Rearranging the ideal gas law allows for the derivation of equations related to pressure, volume, moles, and temperature, particularly when dealing with two sets of values. This is essential in problems involving variables such as two pressures, two temperatures, or two volumes. Understanding these derivations is crucial for applying the ideal gas law effectively in various scenarios, ensuring accurate calculations in thermodynamics and gas behavior.

Ideal Gas Law Derivations

The ideal gas law, represented by the equation PV = nRT, is a fundamental principle in chemistry that relates pressure (P), volume (V), number of moles (n), the ideal gas constant (R), and temperature (T). When faced with problems involving two sets of different values for these variables, we can rearrange the ideal gas law to derive new equations that help us analyze the situation effectively.

For instance, if we have two pressures (P₁ and P₂), two volumes (V₁ and V₂), or two temperatures (T₁ and T₂), we can utilize the relationships established by the ideal gas law to find unknown values. This approach is particularly useful in scenarios where conditions change, allowing us to maintain the integrity of the gas law while adapting to the new parameters.

In these cases, we can express the relationships as follows:

1. For pressure and volume at constant temperature and moles: P₁V₁ = P₂V₂

2. For temperature and volume at constant pressure and moles: V₁/T₁ = V₂/T₂

3. For pressure and temperature at constant volume and moles: P₁/T₁ = P₂/T₂

These derived equations allow for a systematic approach to solving problems involving gases under varying conditions, ensuring that the ideal gas law remains applicable even when multiple variables are in play. Understanding how to manipulate these relationships is crucial for effectively tackling gas law problems in chemistry.

The Ideal Gas Law Derivations Example

To solve the problem of how temperature affects the volume of a gas, we can utilize the ideal gas law, which is expressed as PV = nRT. In this scenario, we are given two volumes and one temperature, indicating that we need to derive a new formula based on the ideal gas law.

Since the pressure remains constant, we can focus on the relationship between volume and temperature. The relevant variables are the initial volume V_1 and temperature T_1, as well as the final volume V_2 and the unknown final temperature T_2. The derived formula can be expressed as:

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

In this case, we have:

• V_1 = 8.30 \, \text{liters}
• V_2 = 5.25 \, \text{liters}
• T_1 = 202 \, \text{°C} = 202 + 273.15 = 475.15 \, \text{K}
• T_2 = ?

Next, we substitute the known values into the derived formula:

\frac{8.30}{475.15} = \frac{5.25}{T_2}

Cross-multiplying gives us:

8.30 \cdot T_2 = 475.15 \cdot 5.25

Solving for T_2 involves dividing both sides by 8.30:

T_2 = \frac{475.15 \cdot 5.25}{8.30} \approx 300.55 \, \text{K}

Finally, to convert T_2 back to degrees Celsius, we subtract 273.15:

T_2 \approx 300.55 - 273.15 \approx 27.40 \, \text{°C}

Thus, the temperature needed to decrease the volume of sulfur hexachloride gas to 5.25 liters is approximately 27.40 degrees Celsius.

PRACTICE: A sample of nitrogen dioxide gas at 130 ºC and 315 torr occupies a volume of 1750 mL. What will the gas pressure be if the volume is reduced to 320 mL at 130 ºC?

• A. 0.65 atm
• B. 0.83 atm
• C. 0.92 atm
• D. 1.6 atm
• E. 2.3 atm

Answer: (E)

A cylinder with a movable piston contains 0.615 moles of gas and has a volume of 295 mL. What will its volume be if 0.103 moles of gas escaped?

• A. 0.176 L
• B. 0.217 L
• C. 0.246 L
• D. 0.361 L
• E. 1.28 L

Answer: (C)

On most spray cans it is advised to never expose them to fire. A spray can is used until all that remains is the propellant gas, which has a pressure of 1350 torr at 25 ºC. If the can is then thrown into a fire at 455 ºC, what will be the pressure (in torr) in the can?

• A. 750 torr
• B. 1800 torr
• C. 2190 torr
• D. 2850 torr
• E. 3300 torr

Answer: (E)

Here’s what students ask on this topic:

What is the derivation of the combined gas law from the ideal gas law?

The combined gas law is derived from the ideal gas law, $P_1{V}_1/T_1=P_2{V}_2/T_2$, assuming the number of moles remains constant. By rearranging the ideal gas law, $PV=nRT$, and dividing the equation for two states of the gas, we eliminate $n$ and $R$ since they are constants. This results in the combined gas law, which relates pressure, volume, and temperature for a gas undergoing changes. It is useful for solving problems where multiple variables change simultaneously.

How do you derive Boyle's law from the ideal gas law?

Boyle's law states that pressure and volume are inversely proportional at constant temperature and moles of gas. Starting with the ideal gas law, $PV=nRT$, we hold $n$, $R$, and $T$ constant. This simplifies the equation to $PV=k$, where $k$ is a constant. Rearranging gives $P=k/V$, showing the inverse relationship between pressure and volume.

How can the ideal gas law be used to derive Charles's law?

Charles's law states that volume is directly proportional to temperature at constant pressure and moles of gas. Using the ideal gas law, $PV=nRT$, we hold $P$, $n$, and $R$ constant. This simplifies to $V/T=k$, where $k$ is a constant. Rearranging gives $V=kT$, demonstrating the direct relationship between volume and temperature.

What is the derivation of Gay-Lussac's law from the ideal gas law?

Gay-Lussac's law states that pressure is directly proportional to temperature at constant volume and moles of gas. Starting with the ideal gas law, $PV=nRT$, we hold $V$, $n$, and $R$ constant. This simplifies to $P/T=k$, where $k$ is a constant. Rearranging gives $P=kT$, showing the direct relationship between pressure and temperature.

How do you derive Avogadro's law from the ideal gas law?

Avogadro's law states that volume is directly proportional to the number of moles of gas at constant pressure and temperature. Using the ideal gas law, $PV=nRT$, we hold $P$, $R$, and $T$ constant. This simplifies to $V/n=k$, where $k$ is a constant. Rearranging gives $V=kn$, demonstrating the direct relationship between volume and moles.