Chemistry Gas Laws: Combined Gas Law - Online Tutor, Practice Problems & Exam Prep

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

Now we have the combined gas law. The combined gas law was created from combining Boyle's law, Charles' law, and Gay-Lussac's law. We're going to say it highlights the relationship between the variables of pressure, volume, and temperature. Remember, Boyle's law states that pressure is inversely proportional to volume. Charles' law states that volume is directly proportional to temperature, and Gay-Lussac's law states that pressure is directly proportional to temperature.

If we take a look down here, we can figure out how the combined gas law was derived. So it comes from combining Boyle's law, Charles' law, and Gay-Lussac's law. If we look, we have pressure and volume as numerators. Volume here is a numerator for Charles' law and Gay-Lussac's law have as their denominators, temperature. So that comes down here.

So here we'd say that the combined gas law is pvt = k, where k would be a constant. This here would represent our combined gas law.

And if we're dealing with 2 sets of pressures, volumes, and temperatures, then it can go further and say p1v1t1 = p2v2t2. So this is how the combined gas law can be derived from these earlier chemistry gas laws.

Chemistry Gas Laws: Combined Gas Law Example 1

Here in this example question, it says a sample of gas initially has a volume of 900 milliliters at 520 kelvin and 1.85 atmospheres. What is the pressure of the gas if the volume decreases to 330 milliliters while the temperature increases to 770 kelvin. Alright. So in this question, they're giving me a volume. We're going to say it's V₁, because later, they give me a second new volume, V₂. They give me a temperature T₁, and then later give me a second temperature, T₂. They give me this pressure in atmospheres, so this is P₁ because later they ask me what the new pressure is. So they're asking us to find P₂.

Now we have our ideal gas law PV = nRT. If you have watched my videos on ideal gas law derivations, you know that we can manipulate the ideal gas law to get our combined gas law. Now if you haven't watched those videos, I highly suggest you go back and take a look. So, look under ideal gas law derivation videos. Now here we're talking about 2 pressures, 2 volumes, 2 temperatures. Moles aren't being discussed because they're being held constant. R is a constant. We divide out the R so that everything is on the left side. And we see that since we're dealing with 2 different values for these variables, it becomes P₁V₁/T₁ = P₂V₂/T₂.

So here we just showed how we derived the combined gas law. Alright. So now we're going to plug in the values that we have. Our pressure is 1.85 atmospheres initially. Our volume when we change milliliters to liters gives us 0.900 liters. Remember, the temperature must always be in kelvins when doing calculations. It's already in kelvins, so we don't have to worry about that.

Now for the calculation, P₂ = (P₁ ⋅ V₁ ⋅ T₂)/(V₂ ⋅ T₁) = (1.85 atm ⋅ 0.900 L ⋅ 770 K) / (0.330 L ⋅ 520 K) = 7.47 atm. So we can see that by changing our volume and changing our temperature, it's caused this change in our pressure too, which comes out again to 7.47 atmospheres.

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

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

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:

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