Chemistry Gas Laws: Combined Gas Law: Videos & Practice Problems

The combined gas law integrates Boyle's law, Charles's law, and Gay Lussac's law, illustrating the relationship between pressure, volume, and temperature. Boyle's law indicates that pressure is inversely proportional to volume, while Charles's law shows that volume is directly proportional to temperature. Gay Lussac's law states that pressure is directly proportional to temperature. The combined gas law can be expressed as ${\mathrm{PV}}^T=k$, where k is a constant, allowing for calculations involving two sets of conditions: $P_1{V}_1/T_1=P_2{V}_2/T_2$.

Chemistry Gas Laws: Combined Gas Law

The combined gas law is a fundamental principle in chemistry that illustrates the relationship between pressure, volume, and temperature of a gas. It is derived from three key gas laws: Boyle's law, Charles' law, and Gay-Lussac's law. Each of these laws describes how one variable affects another under specific conditions.

Boyle's law states that pressure (P) is inversely proportional to volume (V) when temperature is held constant, which can be expressed mathematically as:

P \propto \frac{1}{V}

Charles' law indicates that volume is directly proportional to temperature (T) when pressure is constant:

V \propto T

Gay-Lussac's law asserts that pressure is directly proportional to temperature when volume is constant:

P \propto T

By combining these relationships, we arrive at the combined gas law, which can be expressed as:

\frac{PV}{T} = k

Here, k represents a constant that remains unchanged for a given amount of gas. This equation allows us to analyze the behavior of a gas when two sets of conditions are known. For example, if we have two sets of pressures, volumes, and temperatures, the law can be rearranged to:

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

This formulation is particularly useful in solving problems related to gas behavior, enabling predictions about how changes in one variable will affect the others. Understanding the combined gas law is essential for mastering concepts in thermodynamics and physical chemistry.

Chemistry Gas Laws: Combined Gas Law Example 1

In this scenario, we are analyzing the behavior of a gas sample as its volume and temperature change, while the number of moles remains constant. Initially, the gas has a volume of 900 milliliters (0.900 liters), a temperature of 520 Kelvin, and a pressure of 1.85 atmospheres. When the volume decreases to 330 milliliters (0.330 liters) and the temperature increases to 770 Kelvin, we need to determine the new pressure, denoted as \( P_2 \).

To solve this problem, we utilize the combined gas law, which is derived from the ideal gas law \( PV = nRT \). Since the number of moles (n) and the gas constant (R) remain unchanged, we can express the relationship between the initial and final states of the gas as:

\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

Here, \( P_1 \) is the initial pressure, \( V_1 \) is the initial volume, \( T_1 \) is the initial temperature, \( P_2 \) is the final pressure, \( V_2 \) is the final volume, and \( T_2 \) is the final temperature. Plugging in the known values:

\[ \frac{1.85 \, \text{atm} \times 0.900 \, \text{L}}{520 \, \text{K}} = \frac{P_2 \times 0.330 \, \text{L}}{770 \, \text{K}} \]

To isolate \( P_2 \), we can cross-multiply:

\[ P_2 = \frac{1.85 \, \text{atm} \times 0.900 \, \text{L} \times 770 \, \text{K}}{0.330 \, \text{L} \times 520 \, \text{K}} \]

After performing the calculations, we find that:

\[ P_2 \approx 7.47 \, \text{atm} \]

This result indicates that the pressure of the gas increases significantly due to the decrease in volume and the increase in temperature, demonstrating the direct relationship between these variables as described by the combined gas law.

A 4.30 L gas has a pressure of 7.0 atm when the temperature is 60.0 ºC. What will be the temperature of the gas mixture if the volume and pressure are decreased to 2.45 L and 403.0 kPa respectively?

• A. 108 K
• B. 181 K
• C. 206 K
• D. 310 K
• E. 381 K

Answer: (A)

A sealed container with a movable piston contains a gas with a pressure of 1380 torr, a volume of 820 mL and a temperature of 31°C. What would the volume be if the new pressure is now 2.83 atm, while the temperature decreased to 25°C?

• A. 0.0253 L
• B. 0.167 L
• C. 0.326 L
• D. 0.516 L
• E. 1.46 L

Answer: (D)

Here’s what students ask on this topic:

What is the combined gas law and how is it derived?

The combined gas law integrates Boyle's law, Charles's law, and Gay Lussac's law to describe the relationship between pressure, volume, and temperature of a gas. Boyle's law states that pressure is inversely proportional to volume (P ∝ 1/V), Charles's law states that volume is directly proportional to temperature (V ∝ T), and Gay Lussac's law states that pressure is directly proportional to temperature (P ∝ T). By combining these relationships, the combined gas law is derived as:

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

where k is a constant. For two sets of conditions, it can be expressed as:

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

How do you use the combined gas law to solve problems?

To use the combined gas law to solve problems, follow these steps:

1. Identify the initial and final conditions of pressure (P), volume (V), and temperature (T).
2. Ensure that the temperature is in Kelvin (K). Convert if necessary using T(K) = T(°C) + 273.15.
3. Use the combined gas law formula:

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$

4. Plug in the known values and solve for the unknown variable.

For example, if you know P₁, V₁, T₁, P₂, and T₂, you can solve for V₂ by rearranging the equation:

$V_2=\frac{P_1{V}_1{T}_2}{P_2{T}_1}$

What are some real-life applications of the combined gas law?

The combined gas law has several real-life applications, including:

• Weather Balloons: As a weather balloon ascends, the pressure decreases and the temperature changes, affecting the volume of the balloon. The combined gas law helps predict these changes.
• Refrigeration: The combined gas law is used to understand the behavior of refrigerants as they undergo pressure, volume, and temperature changes in cooling systems.
• Scuba Diving: Divers use the combined gas law to calculate how the pressure and volume of air in their tanks will change with depth and temperature.
• Automobile Engines: The combined gas law helps in understanding the behavior of gases in internal combustion engines, where pressure, volume, and temperature changes occur rapidly.

Why is it important to use Kelvin for temperature in the combined gas law?

It is important to use Kelvin for temperature in the combined gas law because the Kelvin scale is an absolute temperature scale. The combined gas law is derived from the ideal gas law, which requires absolute temperatures to maintain proportionality. Using Celsius or Fahrenheit can lead to incorrect results because these scales do not start at absolute zero, where the kinetic energy of particles theoretically stops. The Kelvin scale ensures that temperature values are always positive and proportional to the energy of the gas particles, making calculations accurate and consistent.

How does the combined gas law relate to the ideal gas law?

The combined gas law is a specific case of the ideal gas law. The ideal gas law is expressed as:

$PV=nRT$

where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature in Kelvin. The combined gas law can be derived from the ideal gas law by considering a fixed amount of gas (n is constant) and combining the relationships of Boyle's law, Charles's law, and Gay Lussac's law. The combined gas law is useful for comparing two sets of conditions without needing to know the number of moles of gas:

$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$