Chemistry Gas Laws: Combined Gas Law: Study with Video Lessons, Practice Problems & Examples

The combined gas law integrates Boyle's law, Charles' 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' 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.

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 says that pressure is inversely proportional to volume. Charles' law says that volume is directly proportional to temperature, and Gay-Lussac's law says 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. So volume here, numerator, pressure Charles' law, and Gay-Lussac's law have as their denominators, temperature. So that comes down here. Here we'd say that the combined gas law is p⁢VT = k, where k here would be a constant. This here would represent our combined gas law. And if we're dealing with two 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 on 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 in order 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 derivation videos. Now here we're talking about 2 pressures, 2 volumes, and 2 temperatures. Moles aren't being discussed because they're being held constant. R is a constant. We divide out the T 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 P1V1T1 = P2V2T2. 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, temperature must always be in kelvins when doing calculations. It's already in kelvins, so we don't have to worry about that. We don't know what P₂ is, V₂ is 0.330 liters, and then temperature T₂ is 770 Kelvin. We're just looking to isolate P₂, so you can cross-multiply these, and then you can cross-multiply these, so that you can isolate P₂. So if I come over here, I'm going to see that, P2 = 1.85 atmospheres × 0.900 liters × 770 kelvins0.330 liters × 520 kelvins.

So we divide out the volume and the temperature that we have here on both sides. So dividing both sides by that, we have isolated at the end our P₂. So liters and kelvins cancel out, and we'll see here that P₂ equals 7.47 atmospheres. So we can see that by changing our volume and changing our temperature, it has 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' law, and Gay-Lussac's law to describe the relationship between pressure (P), volume (V), and temperature (T). Boyle's law states that pressure is inversely proportional to volume (P ∝ 1/V), Charles' 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, the law 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 known and unknown variables (P1, V1, T1, P2, V2, T2).

2. Ensure all temperatures are in Kelvin (K = °C + 273.15).

3. Use the combined gas law equation:

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

4. Rearrange the equation to solve for the unknown variable.

5. Substitute the known values into the equation and solve.

For example, if you need to find V2, rearrange the equation to:

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

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

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

1. **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.

2. **Refrigeration**: In refrigeration systems, the combined gas law helps in understanding how changes in pressure and temperature affect the volume of refrigerants.

3. **Diving**: Scuba divers use the combined gas law to understand how pressure changes with depth affect the volume of air in their tanks and lungs.

4. **Aerospace**: Engineers use the combined gas law to design and test the behavior of gases in different atmospheric conditions, crucial for aircraft and spacecraft design.

Why is it important to convert temperatures to Kelvin when using the combined gas law?

It is important to convert temperatures to Kelvin when using the combined gas law because the Kelvin scale is an absolute temperature scale. The combined gas law is derived based on the absolute temperature, which starts at absolute zero (0 K), where molecular motion theoretically ceases. Using Celsius or Fahrenheit can lead to incorrect results because these scales do not start at absolute zero. The Kelvin scale ensures that the proportional relationships between pressure, volume, and temperature are maintained accurately. To convert Celsius to Kelvin, use the formula:

$K=\mathrm{°C}+\; 273.15$

How does the combined gas law relate to Boyle's, Charles', and Gay-Lussac's laws?

The combined gas law is a synthesis of Boyle's, Charles', and Gay-Lussac's laws, each describing a specific relationship between pressure, volume, and temperature:

1. **Boyle's Law**: States that pressure is inversely proportional to volume at constant temperature (P ∝ 1/V).

2. **Charles' Law**: States that volume is directly proportional to temperature at constant pressure (V ∝ T).

3. **Gay-Lussac's Law**: States that pressure is directly proportional to temperature at constant volume (P ∝ T).

The combined gas law integrates these relationships into a single equation:

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

This equation allows for calculations involving changes in pressure, volume, and temperature simultaneously.