12.3: The Combined Gas Law

Learning Objectives

• To predict the properties of gases using the gas laws.

In the previous section you read about the ideal gas law which can be used to determine a missing property of a gas when the other three properties are known. This is helpful when one variable cannot be easily measured, such as the pressure inside a container that does not have a pressure gauge. 

In this section we will turn our focus to making predictions about gases that are changing. For instance, what is the new pressure of a gas when the temperature is increased? What is the new volume of a gas when the pressure is decreased? 

A systematic way to learn relationships between properties of gases is to control as many variables as possible and to study how one variable is affected when another changes. For instance, study how the pressure of a gas changes when the temperature is increased or decreased (and other variables such as the amount of gas and volume are kept constant). With four different possible variables (pressure, volume, temperature, and amount of gas) there are several possible relationships to study. Each can be represented by a mathematical equation. 

A more realistic scenario in which multiple properties are changing simultaneously can be represented by a mathematical equation that includes all of the variables that describe a gas. Such an equation is called the combined gas law because it is constructed from a combination of the individual gas laws (mathematical equations) that describe pairs of properties. 

In this section you will learn to use the combined gas law equation, including how to use it as a replacement for all of the individual gas laws that describe how different pairs of properties change individually. 

If the units of similar quantities are not the same, one of them must be converted to the other quantity’s units for the calculation to work out properly. It does not matter which quantity is converted to a different unit; the only thing that matters is that the conversion and subsequent algebra are performed properly. The following example illustrates this process.

Combined Gas Law

Other gas laws can be constructed, but we will focus on one in this section, the combined gas law. 

\[\mathrm{\dfrac{P_1V_1}{n_1T_1}=\dfrac{P_2V_2}{n_2T_2}}\]

Notice that this equation contains copies of each variable (i.e two pressures, two volumes, etc.). The subscript 1 indicates an initial value (before a change); a subscript 2 indicates a final value (after a change). As with the ideal gas law, the temperature must be expressed in Kelvin. For the other variables any relevant unit may be used, but the units must be paired. In other words, both pressures must have the same unit, both volumes must have the same unit, etc. 

Example \(\PageIndex{6}\)

A sample of gas has P₁ = 1.50 atm, V₁ = 10.5 L, and T₁ = 300 K. What is the final volume if P₂ = 0.750 atm and T₂ = 350 K?

Solution

Using the combined gas law, substitute for five of the quantities. Since the amount of gas is not changing, n₁ and n₂ are the same and cancel out. 

\(\mathrm{\dfrac{(1.50\: atm)(10.5\: L)}{300\: K}=\dfrac{(0.750\: atm)(V_2)}{350\: K}}\)

We algebraically rearrange this expression to isolate V₂ on one side of the equation:

\(\mathrm{V_2=\dfrac{(1.50\: atm)(10.5\: L)(350\: K)}{(300\: K)(0.750\: atm)}=24.5\: L}\)

Note how all the units cancel except the unit for volume.

Exercise \(\PageIndex{6}\)

A sample of gas has P₁ = 0.768 atm, V₁ = 10.5 L, and T₁ = 300 K. What is the final pressure if V₂ = 7.85 L and T₂ = 250 K?

Answer

0.856 atm

Example \(\PageIndex{7}\)

A balloon containing a sample of gas has a temperature of 22°C and a pressure of 1.09 atm in an airport in Cleveland. The balloon has a volume of 1,070 mL. The balloon is transported by plane to Denver, where the temperature is 11°C and the pressure is 655 torr. What is the new volume of the balloon?

Solution

The first task is to convert all quantities to the proper and consistent units. The temperatures must be expressed in kelvins, and the pressure units are different so one of the quantities must be converted. Let us convert the atmospheres to torr:

22°C + 273 = 295 K = T₁

11°C + 273 = 284 K = T₂

\(\mathrm{1.09\: atm\times\dfrac{760\: torr}{1\: atm}=828\: torr = P_1}\)

Now we can substitute the quantities into the combined has law:

\(\mathrm{\dfrac{(828\: torr)(1,070\: mL)}{295\: K}=\dfrac{(655\: torr)\times V_f}{284\: K}}\)

To solve for V₂, we multiply the 284 K in the denominator of the right side into the numerator on the left, and we divide 655 torr in the numerator of the right side into the denominator on the left:

\(\mathrm{\dfrac{(828\: torr)(1,070\: mL)(284\: K)}{(295\: K)(655\: torr)}=V_2}\)

Notice that torr and kelvins cancel, as they are found in both the numerator and denominator. The only unit that remains is milliliters, which is a unit of volume. So V₂ = 1,300 mL. The overall change is that the volume of the balloon has increased by 230 mL.

Exercise \(\PageIndex{7}\)

A balloon used to lift weather instruments into the atmosphere contains gas having a volume of 1,150 L on the ground, where the pressure is 0.977 atm and the temperature is 18°C. Aloft, this gas has a pressure of 6.88 torr and a temperature of −15°C. What is the new volume of the gas?

Answer

110,038 L

Direct vs. Inverse Relationships

If only two properties of a gas are changing, it is usually possible to make a qualitative prediction about the size of the unknown variable. This can be very helpful when checking your work!

Direct Relationships

Experiments indicate that as the temperature of a gas sample is increased, its volume increases as long as the pressure and the amount of gas remain constant. Because volume increases as temperature increases, these two properties are directly proportional. If you are solving a problem in which only temperature and volume are changing, then you can check your work to make sure that volume increases when temperature increases and vice versa.

Example \(\PageIndex{4}\): Increasing Temperature

What happens to the volume of a gas if its temperature is decreased? Assume that all other conditions remain constant.

Solution

If the temperature of a gas sample is decreased, the volume decreases as well.

Exercise \(\PageIndex{4}\)

What happens to the temperature of a gas if its volume is increased? Assume that all other conditions remain constant.

Answer

The temperature increases.

Example \(\PageIndex{5}\): Checking Your Work

A gas sample at 20°C has an initial volume of 20.0 L. What is its volume if the temperature is changed to 60°C? Does the answer make sense? Assume that the pressure and the amount of the gas remain constant.

Solution

Although the temperatures are given in degrees Celsius, we must convert them to the kelvins before we can use Charles’s law. Thus,

20°C + 273 = 293 K = T₁ 60°C + 273 = 333 K = T₂

Now we can substitute these values into Charles’s law, along with the initial volume of 20.0 L:

\(\mathrm{\dfrac{20.0\: L}{293\: K}=\dfrac{V_2}{333\: K}}\)

Multiplying the 333 K to the other side of the equation, we see that our temperature units will cancel:

\(\mathrm{\dfrac{(333\: K)(20.0\: L)}{293\: K}=V_2}\)

Solving for the final volume, V₂ = 22.7 L. So, as the temperature is increased, the volume increases. This makes sense because volume is directly proportional to the absolute temperature (as long as the pressure and the amount of the remain constant).

Exercise \(\PageIndex{5}\): Making a Prediction

A gas sample at 35°C has an initial volume of 5.06 L. What is its volume if the temperature is changed to −35°C? First make a prediction: will the new volume be higher or lower than 5.06 L? Then calculate the answer. Assume that the pressure and the amount of the gas remain constant.

Answer

The final volume should be lower than 5.06 L because when temperature decreases volume also decreases. The final volume is 3.91 L.

Inverse Relationships

If the amount of gas in a sample and its temperature are kept constant, then as the pressure of a gas is increased, the volume of the gas decreases proportionately. Because volume decreases as pressure increases, these two properties are inversely proportional. If you are solving a problem in which only pressure and volume are changing, then you can check your work to make sure that volume decreases when pressure increases and vice versa. 

Example \(\PageIndex{1}\): Increasing Pressure in a Gas

What happens to the volume of a gas if its pressure is increased? Assume all other conditions remain the same.

Solution

If the pressure of a gas is increased, the volume decreases in response.

Exercise \(\PageIndex{1}\): Increasing Volume in a Gas

What happens to the pressure of a gas if its volume is increased? Assume all other conditions remain the same.

Answer

If the volume of a gas is increased, the pressure decreases.

Example \(\PageIndex{2}\): Making a Prediction

If a sample of gas has an initial pressure of 1.56 atm and an initial volume of 7.02 L, what is the final volume if the pressure is reduced to 0.987 atm? Assume that the amount and the temperature of the gas remain constant. First predict whether the answer will be larger or smaller than 7.02 L, then perform the calculation to see if you were correct. 

Solution

The key in problems like this is to be able to identify which quantities represent which variables from the relevant equation. The way the question is worded, you should be able to tell that 1.56 atm is P₁, 7.02 L is V₁, and 0.987 atm is P₂. What we are looking for is the final volume—V₂. Therefore, substituting these values into P₁V₁ = P₂V₂: (Because the amount of gas and temperature are not changing n₁ and n₂ cancel out and T₁ and T₂ cancel out)

(1.56 atm)(7.02 L) = (0.987 atm) × V₂

The expression has atmospheres on both sides of the equation, so they cancel algebraically:

(1.56)(7.02 L) = (0.987) × V₂

Now we divide both sides of the expression by 0.987 to isolate V₂, the quantity we are seeking:

\(\mathrm{\dfrac{(1.56)(7.02\: L)}{0.987}=V_2}\)

Performing the multiplication and division, we get the value of V₂, which is 11.1 L. The volume increases. This should make sense because the pressure decreases, so pressure and volume are inversely related.

Exercise \(\PageIndex{2}\)

If a sample of gas has an initial pressure of 3.66 atm and an initial volume of 11.8 L, what is the final pressure if the volume is reduced to 5.09 L? Assume that the amount and the temperature of the gas remain constant.

Answer

8.48 atm

Example \(\PageIndex{3}\): Checking Your Work

If a sample of gas has an initial pressure of 1.56 atm and an initial volume of 7.02 L, what is the final volume if the pressure is changed to 1,775 mm Hg? Does the answer make sense? Assume that the amount and the temperature of the gas remain constant.

Solution

This example is similar to Example \(\PageIndex{2}\), except now the final pressure is expressed in mm Hg. For the math to work out properly, one of the pressure values must be converted to the other unit. Let us change the initial pressure to mm Hg:

\(\mathrm{1.56\: atm\times\dfrac{760\: mm\:Hg}{1\: atm}=1,190\: mm \:Hg}\)

Now we can use Boyle’s law:

(1,190 mm Hg)(7.02 L) = (1,775 mm Hg) × V₂

Torr cancels algebraically from both sides of the equation, leaving

(1,190)(7.02 L) = (1,775) × V₂

Now we divide both sides of the equation by 1,775 to isolate V₂ on one side. Solving for the final volume,

\(\mathrm{V_2=\dfrac{(1,190)(7.02\: L)}{1,775}=4.71\: L}\)

Because the pressure increases, it makes sense that the volume decreases.

The answer for the final volume is essentially the same if we converted the 1,775 mm Hg to atmospheres: \(\mathrm{1,775\: mm\:\times\dfrac{1\: atm}{760\: mm \:Hg}=2.336\: atm}\). Using Boyle’s law: (1.56 atm)(7.02 L) = (2.335 atm) × V₂; \(\mathrm{V_f=\dfrac{(1.56\: atm)(7.02\: L)}{2.336\: atm}=4.69\: L}\).

Exercise \(\PageIndex{3}\)

If a sample of gas has an initial pressure of 375 torr and an initial volume of 7.02 L, what is the final pressure if the volume is changed to 4,577 mL? Does the answer make sense? Assume that amount and the temperature of the gas remain constant.

Answer

575 torr. Since the volume decreased, it makes sense that the pressure increased because pressure and volume are inversely related. 

To Your Health: Breathing

Breathing certainly is a major contribution to your health! Without breathing, we could not survive. Curiously, the act of breathing itself is little more than an application of the inverse relationship between volume and pressure (called Boyle’s law).

The lungs are a series of ever-narrowing tubes that end in a myriad of tiny sacs called alveoli. It is in the alveoli that oxygen from the air transfers to the bloodstream and carbon dioxide from the bloodstream transfers to the lungs for exhalation. For air to move in and out of the lungs, the pressure inside the lungs must change, forcing the lungs to change volume—just as predicted by Boyle’s law.

The pressure change is caused by the diaphragm, a muscle that covers the bottom of the lungs. When the diaphragm moves down, it expands the size of our lungs. When this happens, the air pressure inside our lungs decreases slightly. This causes new air to rush in, and we inhale. The pressure decrease is slight—only 3 torr, or about 0.4% of an atmosphere. We inhale only 0.5–1.0 L of air per normal breath.

Exhaling air requires that we relax the diaphragm, which pushes against the lungs and slightly decreases the volume of the lungs. This slightly increases the pressure of the air in the lungs, and air is forced out; we exhale. Only 1–2 torr of extra pressure is needed to exhale. So with every breath, our own bodies are performing an experimental test of Boyle’s law.

Determining Relationships

Think about the two relationships that were just discussed: 

• Volume and temperature are directly related: when one increases, the other also increases
• Pressure and volume are inversely related: when one increases, the other decreases

Now take a look at the combined gas law and see if you can figure out whether relationships between other variables (e.g. pressure and temperature) will be direct or inverse. 

\[\mathrm{\dfrac{P_1V_1}{n_1T_1}=\dfrac{P_2V_2}{n_2T_2}}\]

You can determine whether a pair of properties are directly or inversely related by looking at their relative positions in the mathematical equation. If two variables must be multiplied to obtain a constant, like P₁ and V₁, then they are inversely proportional. If two variables must be divided to obtain a constant, like V₁ and T₁, then they are directly proportional. 

Concept Review Exercises

1. What properties do the gas laws help us predict?
2. What makes the ideal gas law different from the other gas laws?

Answers

1. Gas laws relate four properties: pressure, volume, temperature, and number of moles.
2. The ideal gas law does not require that the properties of a gas change.

Key Takeaway

• The physical properties of gases are predictable using mathematical formulas known as gas laws.

Exercises

1. What conditions of a gas sample should remain constant for Boyle’s law to be used?

2. What conditions of a gas sample should remain constant for Charles’s law to be used?

3. Does the identity of a gas matter when using Boyle’s law? Why or why not?

4. Does the identity of a gas matter when using Charles’s law? Why or why not?

5. A sample of nitrogen gas is confined to a balloon that has a volume of 1.88 L and a pressure of 1.334 atm. What will be the volume of the balloon if the pressure is changed to 0.662 atm? Assume that the temperature and the amount of the gas remain constant.

6. A sample of helium gas in a piston has a volume of 86.4 mL under a pressure of 447 torr. What will be the volume of the helium if the pressure on the piston is increased to 1,240 torr? Assume that the temperature and the amount of the gas remain constant.

7. If a gas has an initial pressure of 24,650 Pa and an initial volume of 376 mL, what is the final volume if the pressure of the gas is changed to 775 torr? Assume that the amount and the temperature of the gas remain constant.

8. A gas sample has an initial volume of 0.9550 L and an initial pressure of 564.5 torr. What would the final pressure of the gas be if the volume is changed to 587.0 mL? Assume that the amount and the temperature of the gas remain constant.

9. A person draws a normal breath of about 1.00 L. If the initial temperature of the air is 18°C and the air warms to 37°C, what is the new volume of the air? Assume that the pressure and amount of the gas remain constant.

10. A person draws a normal breath of about 1.00 L. If the initial temperature of the air is −10°C and the air warms to 37°C, what is the new volume of the air? Assume that the pressure and the amount of the gas remain constant.

11. An air/gas vapor mix in an automobile cylinder has an initial temperature of 450 K and a volume of 12.7 cm³. The gas mix is heated to 565°C. If pressure and amount are held constant, what is the final volume of the gas in cubic centimeters?

12. Given the following conditions for a gas: V_i = 0.665 L, T_i = 23.6°C, V_f = 1.034 L. What is T_f in degrees Celsius and kelvins?

13. Assuming the amount remains the same, what must be the final volume of a gas that has an initial volume of 387 mL, an initial pressure of 456 torr, an initial temperature of 65.0°C, a final pressure of 1.00 atm, and a final temperature of 300 K?

14. When the nozzle of a spray can is depressed, 0.15 mL of gas expands to 0.44 mL, and its pressure drops from 788 torr to 1.00 atm. If the initial temperature of the gas is 22.0°C, what is the final temperature of the gas?

15. Use the ideal gas law to show that 1 mol of a gas at STP has a volume of about 22.4 L.

16. Use a standard conversion factor to determine a value of the ideal gas law constant R that has units of L•torr/mol•K.

17. How many moles of gas are there in a 27.6 L sample at 298 K and a pressure of 1.44 atm?

18. How many moles of gas are there in a 0.066 L sample at 298 K and a pressure of 0.154 atm?

19. A 0.334 mol sample of carbon dioxide gas is confined to a volume of 20.0 L and has a pressure of 0.555 atm. What is the temperature of the carbon dioxide in kelvins and degrees Celsius?

20. What must V be for a gas sample if n = 4.55 mol, P = 7.32 atm, and T = 285 K?

21. What is the pressure of 0.0456 mol of Ne gas contained in a 7.50 L volume at 29°C?

22. What is the pressure of 1.00 mol of Ar gas that has a volume of 843.0 mL and a temperature of −86.0°C?

Answers

1. temperature and amount of the gas

2. pressure and amount of the gas

3. The identity does not matter because the variables of Boyle’s law do not identify the gas.

4. The identity does not matter because the variables of Charles law do not identify the gas.

5. 3.79 L

6. 31.1 mL

7. 92.1 mL

8. 918.4 torr

9. 1.07 L

10. 1.18 L

11. 23.7 cm³

12. 461 K; 188⁰C

13. 206 mL

14. 835 K; 562⁰C

15. The ideal gas law confirms that 22.4 L equals 1 mol.

16. \(\dfrac{760\: torr}{1\: atm}\)

17. 1.63 mol

18. 4.2 x 10^-4 mol

19. 405 K; 132°C

20. 14.5 L

21. 0.151 atm

22. 18.2 atm