The ideal gas laws show the correlation of the temperature, pressure, volume, and the amount of a gas. It is derived from the individual gas laws such as the Boyle’s law, Charles’ law, etc. Remember, when discussing those, we used a pump with a freely moving plunger and by changing the parameters one by one, we were able to get the following correlations:

We can combine all of these to get an expression showing how, for example, the volume of an ideal gas is correlated to the temperature, pressure, and the moles of the gas:

To bring in the equal sign, we introduce a constant:

This can be explained using the example of a car dealership income. The income depends on the number of sales which we can represent as:

income **~** number of cars sold

However, we cannot say income = number of cars sold, so to switch an equal sign, we need to introduce a constant. This can be the price of the car transforming the equation to:

Income = price x number of cars

Now, for the parameters of ideal gases, the constant is the *R*, which is called the ideal gas constant.

\[V\;{\rm{ = }}\,R\, \times \,\frac{{nT}}{P}\]

By rearranging it, we obtain the **ideal gas law equation**:

Again, *R* is called the ideal gas constant. Although it has different values and units, you will mostly be using this:

\[R\;{\rm{ = }}\;{\rm{0}}{\rm{.08206}}\;\frac{{{\rm{L}} \cdot {\rm{atm}}}}{{{\rm{mol}} \cdot {\rm{K}}}}\]

**The Ideal Gas Law in Problems**

The ideal gas law equation is used when you need to find ** P, V, T, **or

**, for the system where they**

*n***do not change**.

**For example**, A sample of hydrogen gas is added into a 5.80 L container at 56.0 °C. How many moles of the gas are present in the container if the pressure is 6.70 atm?

Rearrange the ideal gas law to get an expression for the moles (n):

PV = nRT

\[{\rm{n}}\;{\rm{ = }}\;\frac{{{\rm{PV}}}}{{{\rm{RT}}}}\]

Now, before entering the numbers, remember to always convert the temperature to Kelvin whenever solving a gas problem:

T = 56.0 oC + 273 = 329 K

And now, we can plug in the numbers:

\[{\rm{n}}\;{\rm{ = }}\;\frac{{{\rm{6}}{\rm{.70}}\;\cancel{{{\rm{atm}}}}\; \times \;{\rm{5}}{\rm{.80}}\;\cancel{{\rm{L}}}}}{{{\rm{0}}{\rm{.08206}}\;\cancel{{\rm{L}}}\;\cancel{{{\rm{atm}}}}\;\cancel{{{{\rm{K}}^{{\rm{ – 1}}}}}}\;{\rm{mo}}{{\rm{l}}^{{\rm{ – 1}}}}{\rm{ }} \times \;{\rm{329}}\;\cancel{{\rm{K}}}}}\; = \;1.44\;{\rm{mol}}\]

**Molar Mass and Density of Gases**

Density and molar mass are calculated by modifying the ideal gas law equation. What we do is first replace the moles (n) in PV = nRT with mass over the molar mass:

Next, we rearrange this formula to bring the V under the m which is the density of the gas (mass/volume = density):

From this equation, we can obtain an expression for the density and molar mass of an ideal gas:

Check this article for details and some examples of determining the molar and the density using the ideal gas law equation.

** **

**How do I know which gas law to use?**

When the problem has varying parameters, then instead of trying to figure out which gas law to use, you can go with the combined gas law equation:

** **

** **

** **

Let’s keep it for another article because there is quite a lot of information in this one.

Here are some summary of gas law problems and if you are not sure how to start, make sure to check the articles on the gas law chapter.

#### Practice

The pressure of a gas is 2.30 atm in a 1.80 L container. Calculate the final pressure of the gas if the volume is decreased to 1.20 liters.

After changing the pressure of a gas sample from 760.0 torr to 0.800 atm, it occupies 4.30 L volume. What was the initial volume of the gas?

What will be the final volume of a 3.50 L sample of nitrogen at 20.0 °C if it is heated to 200. °C?

The volume of a gas decreased from 2.40 L to 830. mL and the final temperature is set at 40.0 °C. Assuming a constant pressure, calculate the initial temperature of the gas in kelvins.

A sample of helium gas at 1.40 atm is heated from 23.0 °C to 400.0 K. How many atmospheres is the final pressure of the helium gas?

A sample of hydrogen gas is added in a 5.80 L container at 56.0 °C. How many moles of the gas are present in the container if the pressure is 6.70 atm?

What is the pressure in a 26.0 L container with 5.40 moles of nitrogen dioxide if the temperature is 64.0°C?

A 3.7 L gas sample, initially at STP, is heated to 280. °C at constant volume. Calculate the final pressure of the gas in atm.

A 2.65 g sample of dry ice (solid carbon dioxide) is placed in a 2.90-L vessel and converted into CO_{2} gas. Calculate the pressure inside the vessel if the temperature is at 35.0 ^{o}C.

A gas sample is stored in a 429 mL container at 9.50°C and 2.20 atm. Calculate the pressure of the gas if the volume changes to 134 mL and the container is heated to 134.5°C? Assume a constant amount of gas.

A gas sample occupies 22.0 L at 171°C and 1.43 atm. Calculate the volume of the gas if its temperature and pressure are increased to 197 °C and 1.80 atm respectively.

A sample of ethylene gas (C_{2}H_{6}) collected in a 36.4 mL vessel with a freely moving piston, at 31.0 °C exerts 745 mmHg pressure. What is the volume of this gas at STP?

A gas-filled balloon having a volume of 3.50 L at 1.30 atm and 25.0 °C is allowed to rise 5 km above the surface of Earth, where the temperature and pressure are 12.0 °C and 1.10 atm, respectively. What would the volume of the balloon be in these conditions?

What is the density of CO_{2} gas at 386 K and 17.0 atm.

Determine the density of ammonia gas, NH_{3}, at 36.0 ^{o}C and 695 mmHg. Report the density in grams per liter.

A scientist carries out an experiment to determine the molar mass of a 2.84-g sample of a colorless liquid which exerts 756 mmHg pressure when vaporized in a 260-mL flask at 142 ^{o}C. What is the molecular mass of this compound?

Identify the unknown gas that weighs 17.75 grams in a 17.0 L cylinder held at 0.700 atm pressure and 250°C.

a) NO_{2} b) CO_{2} c) H_{2} d) SO_{2} e) He

What is the ratio of the effusion rates of hydrogen gas (H_{2}) and carbon dioxide (CO_{2}) at the same pressure and temperature?

The rate of effusion of an unknown gas is 9.20 mL/min. Under identical conditions, the rate of effusion of pure nitrogen (N_{2}) gas is 14.65 mL/min. Identify the unknown gas using the Graham’s law.

a) O_{2} b) C_{3}H_{8} c) C_{4}H_{10} d) NO_{2} e) Cl_{2}

A sample of krypton effuses from a container in 95 seconds. The same amount of an unknown gas requires 55 seconds. Identify the unknown gas.

If a sample of Br_{2} vapor can effuse from an opening in a heated vessel in 46 s, how long will it take the same amount of He to effuse under identical conditions?

It has been demonstrated that 3.56 mL of an unknown gas effuses through a hole in the same time that 8.64 mL of argon does under the same conditions. Determine the molecular mass of the unknown gas.

What is the partial pressure of nitrogen in a mixture of N_{2}, SO_{2} and CO_{2} that has a total pressure of 6.84 atm. The partial pressure of SO_{2} is 2.10 atm, and the partial pressure of CO_{2 }is 1.74 atm.

13.2 grams of CO_{2 }and 6.00 grams of He are mixed in a 4.00 L container at 300. K. Calculate the partial pressure of both gases and the total pressure of the mixture.

A 3.00-L bulb containing N_{2} at 1.80 atm pressure is connected to a 2.00-L bulb filled with H_{2} at 3.50 atm pressure. What is the final pressure of the system when the valve is opened?

Assuming ideal gas behavior, calculate the total pressure (in atm) of a mixture of 0.0260 mol of nitrogen, N_{2}, and 0.0170 mol of argon, Ar, in a 3.50-L flask at 20 ^{o}C.

A mixture of gases contains CH_{4}, N_{2}, and H_{2} and exerts a total pressure of 2.65 atm. The mixture contains 0.456 mol of CH_{4}, 0.540 mol of N_{2} and 0.730 mol of H_{2}. What is the partial pressure of hydrogen in atmospheres?

The partial pressures of CH_{4}, C_{3}H_{8}, and C_{4}H_{10} in a gas mixture are 270 torr, 1016 torr, and 1142 torr, respectively. What is the mole fraction of butane (C_{4}H_{10})?

In a mixture of two gases, the partial pressure of CO_{2}(*g*) is 0.145 atm and that of O_{2}(*g*) is 0.370 atm.

a) What is the mole fraction of each gas in the mixture?

b) Calculate the total number of moles of gas in the mixture if the mixture occupies a volume of 12.5 L at 45.0 °C.

c) Calculate the number of grams of each gas in the mixture.

Oxygen can be prepared in the laboratory by decomposing potassium chlorate, KClO_{3} according to the following reaction:

2KClO_{3}(s) → 2KCl(s) + 3O_{2}(g).

How many liters of oxygen can be produced from 4.80 g KClO_{3}, if the reaction is carried out at standard conditions?

Sodium azide is used in automobile airbags as a source of nitrogen gas to rapidly inflate the bags upon a hit. If the volume of the airbag is 7.95 L, what mass of NaN_{3} is required to produce enough nitrogen to fill it at 23.0 °C and 1.20 atm?

2NaN_{3}(*s*) → 2Na(*s*) + 3N_{2}(*g*)

Calcium hydride (CaH_{2}) can be used as a drying agent and a source of Hydrogen, because of its high reactivity with water.

CaH_{2}(*s*) + 2 H_{2}O(*l*) → Ca(OH)_{2}(*aq*) + 2H_{2}(*g*)

How many grams of CaH_{2} are needed to produce 76.8 L of H_{2} gas at a pressure of 0.750 atm and a temperature of 25 °C?

Propane (C_{3}H_{8}) is used as a fuel in gas barbecue grills. How many liters of oxygen, taken at STP, are needed for the full combusting of 26.4 g of propane?

C_{3}H_{8}(*g*) + 5O_{2}(*g*) → 3CO_{2}(*g*) + 4H_{2}O(*l*)

Lithium hydroxide (LiOH) is used in space shuttles to absorb the carbon dioxide exhaled by astronauts according to the following equation:

2LiOH(*s*) + CO_{2}(*g*) → Li_{2}CO_{3}(*s*) + H_{2}O(*l*)

How many liters of carbon dioxide gas at 23.0 ^{o}C and 732 mmHg could be absorbed by 284 g of lithium hydroxide?

In a particular reaction, at 25 °C, 16.4 L of carbon monoxide at a 950.0 torr is mixed with 11.2 g of iron (III) oxide, and 5.68 g iron is obtained. What is the percent yield of the reaction?

Fe_{2}O_{3} + 3CO(*g*) → 2Fe + 3CO_{2}(*g*)