Fifteen Examples

The Ideal Gas Law was first written in 1834 by Emil Clapeyron. What follows is just one way to "derive" the Ideal Gas Law. For a static sample of gas, we can write each of the six gas laws as follows:

PV = k_{1}

V / T = k_{2}

P / T = k_{3}

V / n = k_{4}

P / n = k_{5}

1 / nT = 1 / k_{6}

Note that the last law is written in reciprocal form. The subscripts on k indicate that six different values would be obtained.

When you multiply them all together, you get:

P^{3}V^{3}/ n^{3}T^{3}= k_{1}k_{2}k_{3}k_{4}k_{5}/ k_{6}

Let the cube root of k_{1}k_{2}k_{3}k_{4}k_{5} / k_{6} be called R.

The units work out:

k_{1}= L-atm

k_{2}= L / K

k_{3}= atm / K

k_{4}= L / mol

k_{5}= atm / mol

1 / k_{6}= 1 / mol-K

Each unit occurs three times and the cube root yields L-atm / mol-K, the correct units for R when used in a gas law context.

Consequently, we have:

PV / nT = R

or, more commonly:

PV = nRT

R is called the gas constant. Sometimes it is referred to as the universal gas constant. If you wind up taking enough chemistry, you will see it showing up over and over and over.

A second method to "derive" the Ideal Gas Law is to state Boyle's, Charles' and Avogadro's Law as proportions:

1 V ∝ –– P V ∝ T

V ∝ n

Therefore:

nT V ∝ ––– P

Introduce a proportionality constant and name it R:

nRT V = ––– P

One minor rearrangement to get the most common manner of writing the law:

PV = nRT

By the way, this same technique can be used with other combinations of three of the six two-variable gas laws. You just need to have the same value be on the left-hand side of each proportion. Try Boyle, Gay-Lussac, and Diver's and focus on pressure being on the left-hand side.

R's value can be determined many ways. This is just one way:

We will assume we have 1.000 mol of a gas at STP. The volume of this amount of gas under the conditions of STP is known to a high degree of precision. We will use the value of 22.414 L.

By the way, 22.414 L at STP has a name. It is called molar volume. It is the volume of ANY ideal gas at standard temperature and pressure.

Let's plug our numbers into the equation:

(1.000 atm) (22.414 L) = (1.000 mol) (R) (273.15 K)

Notice how atmospheres were used as well as the exact value for standard temperature.

Solving for R gives 0.08206 L atm / mol K, when rounded to four significant figures. This is usually enough. Remember the value. You'll need it for problem solving.

Notice the weird unit on R: say out loud "liter atmospheres per mole Kelvin."

This is not the only value of R that can exist. (Here's a whole bunch.) It depends on which units you select. Those of you that take more chemistry than high school level will meet up with 8.31447 Joules per mole Kelvin, but that's for another time. The ChemTeam will use the 0.08206 value in gas-related problems almost every time.

**Example #1:** A sample of gas at 25.0 °C has a volume of 11.0 L and exerts a pressure of 660.0 mmHg. How many moles of gas are in the sample?

**Solution:**

1) PV = nRT:

(660.0 mmHg / 760.0 mmHg/1.00 atm) (11.0 L) = (n) (0.08206 L atm / mol K) (298 K)Note the conversion from mmHg to atm and from Celsius to Kelvin.

2) Solve:

(0.868421 atm) (11.0 L) n = ––––––––––––––––––––––––––– (0.08206 L atm / mol K) (298 K) n = 0.391 mol

**Example #2:** A sample of gas at 28.0 °C has a volume of 6.20 L and exerts a pressure of 720.0 mmHg. How many moles of gas are in the sample?

**Solution:**

1) PV = nRT:

(720.0 mmHg / 760.0 mmHg/1.00 atm) (6.20 L) = (n) (0.08206 L atm / mol K) (301 K)Note the conversion from mmHg to atm and from Celsius to Kelvin.

2) Calculate:

n = 0.238 mol

**Example #3:** Calculate the approximate volume of a 0.400 mol sample of gas at 11.0 °C and a pressure of 2.43 atm.

**Solution:**

1) PV = nRT:

(2.43 atm) (V) = (0.400 mol) (0.08206 L atm / mol K) (284 K)

2) Calculate:

(0.400 mol) (0.08206 L atm / mol K) (284 K) V = ––––––––––––––––––––––––––––––––––––– 2.43 atm V = 3.84 L

**Example #4:** Calculate the approximate temperature of a 0.300 mol sample of gas at 780. mmHg and a volume of 6.00 L.

**Solution:**

1) PV = nRT:

(780 mmHg) (6.00 L) = (0.300 mol) (62.3638 L mmHg / mol K) (284 K)Note the different value and unit for R, to be in agreement with using mmHg for the pressure unit. Alternatively, you could convert 780. mmHg to atm and then use 0.08206 L atm / mol K for the value of R.

A variety of values for R can be found here.

2) Rearrange PV = nRT to isolate T:

T = PV / nRT = [(780.) (6.00)] / [(0.300) (62.3638)]

T = 250. K

The problem does not specify the final unit, but Celsius is most often requested.

250 − 273 = −23 °C

**Example #5:** What is the pressure exerted by 2.3 mol of a gas with a temperature of 40. °C and a volume of 3.5 L?

**Solution:**

PV = nRT(P) (3.5 L) = (2.3 mol) (0.08206 L atm / mol K) (313 K)

P = [(2.3) (0.08206) (313)] / 3.5

P = 16.9 atm

**Example #6:** A sample of dry gas weighing 2.1025 grams is found to occupy 2.850 L at 22.00 °C and 740.0 mmHg. (a) How many moles of the gas are present? (b) What is the molar mass of the gas?

**Solution to (a):**

1) I will change the units for pressure to atm., so as to keep with my preferred value for R:

740.0 mm Hg ÷ 760.0 mm Hg/atm = 0.973684 atm

2) Now, plug into the equation and solve for n:

(0.973684 atm) (2.850 L) = (n) (0.08206 L atm / mol K) (295.0 K)To four sig figs, the answer is 0.1146 mol

**Solution to (b):**

This is a very common use of this law and the odds are very good you will see this type of question on a test.The key is to remember the units on molar mass: grams per mole.

We know from the problem statement that 2.1025 grams of the gas is involved and we know (by the above calculation that 0.1146329 mole of the gas is present.

So all we have to do is divide the grams of gas by how many moles it is:

2.1025 g / 0.1146329 mol = 18.34 g/mol

Let's go over those steps for using the Ideal Gas Law to calculate the molar mass of the gas:

(1) You have to know the grams of gas involved. Usually the problem will just give you the value, but not always. You might have to calculate it.(2) You are going to have to calculate the moles of gas. Use PV = nRT and solve for n. Make sure to use L, atm and K.

(3) Divide grams by moles and there's your answer.

Reminder: it is almost 100% certain that you will be asked a determine-molar-mass question on your Gas Laws test.

**Example #7:** An ideal gas has a density of 1.17 x 10¯^{6} g/cm^{3} at 1.00 x 10¯^{3} atm and 60.0 °C. Identify the gas.

**Solution:**

1) Determine moles of gas in 1.00 L of the sample:

PV = nRT(0.00100 atm) (1.00 L) = (n) (0.08206 L atm/mol K) (333 K)

n = 0.000036595 mol

2) The mass of one liter of gas:

(1.17 x 10¯^{6}g/cm^{3}) (1000 cm^{3}/ 1.00 L) = 1.17 x 10¯^{3}g/L

3) Determine the molecular weight of the gas:

1.17 x 10¯^{3}g / 0.000036595 mol = 31.97 g/molThe molecular weight of oxygen gas (O

_{2}) is 31.9994 g/molThe unknown gas is oxygen.

Hydrazine (N

_{2}H_{4}) weighs a bit more than 32, but it is a liquid at 60 °C

**Example #8:** At STP, a 5.00 L flask filled with air has a mass of 543.251 g. The air in the flask is replaced with another gas and the mass of the flask is 566.107 g. The density of air is 1.29 g/L. What is the gas that replaced the air?

**Solution:**

1) Calculate mass of air in flask:

(1.29 g/L) (5.00 L) = 6.45 g

2) Calculate mass of flask:

543.251 g − 6.45 g = 536.801 g

3) Calculate mass of unknown gas:

566.107 g − 536.801 g = 29.306 g

4) Calculate moles of unknown gas:

PV = nRT(1.00 atm) (5.00 L) = (n) (0.08206) (273 K)

n = 0.22319 mol

5) Calculate molar mass of unknown gas:

29.306 g / 0.22319 mol = 131.3 g/mol

6) Use the periodic table to look up what gas element is closest to 131.3 g/mol. The unknown gas could be xenon.

**Example #9:** A vessel at 25.0 °C is evacuated to a point where the amount of gas remaining is 5.00 x 10^{9} molecules per m^{3}. What is the pressure in the vessel?

**Solution using one cubic meter:**

1) Determine moles of gas present in 1 m^{3}:

(5.00 x 10^{9}molecules) / (6.022 x 10^{23}molecules/mole) = 8.303 x 10¯^{15}mol

2) Use Ideal Gas Law:

PV = nRT(P) (1.00 m

^{3}) = (8.303 x 10¯^{15}mol) (8.20575 x 10¯^{5}m^{3}atm / mol K) (298 K)I looked up the value for R here.

P = 2.03 x 10¯

^{16}atm

Comment by the ChemTeam: I will often change the pressure to atm, so as to use the 0.08206 value. Here, I simply looked up the value of R rather than change the volume to liters. On a test, you may not have that luxury, so consider making sure you are comfortable changing a volume like m^{3} to liters.

**Solution using one liter:**

The ChemTeam did not write this answer, making only some minor style changes.

take 1 liter of volume, that is 0.001 m^{3}. so there are 5e6 molecules.

6.022e23 molecules/mole (Avogadro's constant)

1 mole is 6.022e23 molecules, so

(5e6 molecules) x (1 mol/6.022e23 molecules) = 8.303e-18 molIdeal gas law

PV = nRT

n = number of moles

R = gas constant = 0.08206 (L atm)/(mol K)

T = temperature in Kelvins

P = absolute pressure in atm

V = volume in litersP = nRT/V = (8.303e-18) (0.08206) (298) / 1 = 2.03e-16 atm

**Example #10:** Of the following gases, which has density of 0.906 g/L at 315 K and 1.16 atm.

(a) Ne

(b) Ar

(c) Kr

(d) Xe

(e) He

**Solution #1:**

PV = nRT(1.16 atm) (1.00 L) = (n) (0.08206 L atm / mol K) (315 K)

n = 0.044876 mol

0.906 g / 0.044876 mol = 20.19 g/mol <--- the molar mass of neon

**Solution #2:**

PV = nRTPV = (m/M)RT <--- where m = mass, M = molar mass

Since density = m/V, we have this (after substituting and rearranging):

PVM = mRT

M = mRT / PV

M = [(0.906 g) (0.08206 L atm / mol K) (315 K)] / [(1.00 L) (1.16 atm)] = 20.19 g/mol

Neon.

**Solution #3 (without PV = nRT):**

At STP , 1 mol of gas has a volume of 22.414 LUsing the Combined Gas Law, convert the volume to 315 K and 1.16 atm:

(22.414) (315/273) (1.00/1.16) = 22.295 L

Now, you need to calculate densities until you get the correct answer:

Ne---> 20.180 g / 22.295 L = 0.905 g/L <--- the correct answer

For comparison's sake, here is argon:

Ar ---> 39.95 g / 22.295 L = 1.79 g/L <--- this is a wrong answer

**Example #11:** A 0.105 g sample of an unknown diatomic gas contained in a 125 mL vessel has a pressure of 560 torr at 23 &175;C. What is the molar mass of the gas? What is the identity of the gas?

**Solution:**

PV = nRT(560 torr / 760 torr/atm) (0.125 L) = (n) (0.08206 L atm / mol K) (296 K)

n = 0.003792 mol

0.105 g / 0.003792 mol = 27.7 g/mol

Nitrogen has a molar mass of 28.0 g/mol

**Example #12:** If 1.0 g of each of the following gases is taken at STP, which one would occupy the greatest volume?

(a) CO

(b) H_{2}O

(c) CH_{4}

(d) NO

(e) They would all occupy the same volume.

**Solution:**

1) Modify PV = nRT as follows:

V = nRT / PSince RT/P are the same for each gas, the greatest volume will be for the gas with the greatest number of moles.

2) Consequently, the greatest number of moles in 1.0 g of a gas will be for the compound with the lowest molar mass:

CO ---> 28.0 g/mol

H_{2}O ---> 18.0 g/mol

CH_{4}---> 16.0 g/mol

NO ---> 30.0 g/mol1.0 g of methane, CH

_{4}, will have the greatest volume at STP.

**Example #13:** 12.8 g of liquid helium at 1.7 K is completely vaporized. What volume does the helium occupy at STP?

**Solution:**

We don't care about the 1.7 K. One reason is that, at 1.7 K, the helium is not yet a gas, since helium vaporizes at about 4.25 K. Another reason is that we only care about helium's situation at STP, we don't care what the helium did at lower temperatures.

1) Determine moles of helium:

12.8 g / 4.0026 g/mol = 3.1979 mol

2) Use PV = nRT:

(1.00 atm) (x) = (3.1979 mol) (0.08206 L atm / mol K) (273.15 K)x = 71.6799 L

To three sig figs, 71.7 L

Using the more common 273 K for standard temperature results in 71.6 L. Make sure to use the value for standard temperature that your teacher uses. That is the value he/she uses to calculate the answers on the test.

**Example #14:** An elemental gas (meaning that it is made from one element) has a mass of 10.3 grams. It occupies 58.4 L at 1.00 atm and 2.50 °C. Identify the gas.

**Solution:**

1) Let us use the ideal gas law to determine the moles of gas present:

PV = nRT(1.00 atm) (58.4 L) = (n) (0.08206 L atm mol¯

^{1}K¯^{1}) (275.50 K)n = 2.58321 mol

2) Determine the molecular weight (molar mass) of the gas:

10.3 g / 2.58321 mol = 3.99 g/mol (to three sig figs)

3) The elemental gases are the noble gases, most of the halogens, hydrogen, oxygen, and nitrogen. We examine the atomic weights on the periodic table and determine the gas to be helium.

**Example #15:** An elemental gas (meaning that it is made from one element) has a mass of 10.3 grams. It occupies 4.27 L at 1.00 atm and 85.0 °C. Identify the gas.

**Solution:**

PV = nRT(1.00 atm) (4.27 L) = (n) (0.08206 L atm mol¯

^{1}K¯^{1}) (358.0 K)n = 0.14535 mol

10.3 g / 0.14535 mol = 70.9 g/mol

Look for 70.9 on the periodic table and can't find it. What to do? Aha! Remember that diatomic gases exist. Look for 35.45 (half of 70.9) and find chlorine.

The diatomic gas chlorine, Cl

_{2}, is the answer.

**Bonus Example #1:** One of the methods for estimating the temperature at the center of the sun is based on the ideal gas equation. If the center is assumed to be a mixture of gases whose average molar mass is 2.04 g/mol, and the density and pressure are 1.14 g/cm^{3} and 2.01 x 10^{9} atm, respectively, calculate the temperature.

**Solution:**

1) Use the density and the molar mass to determine the moles of gas in 1.00 L:

1.00 L = 1000 mL = 1000 cm^{3}(1.14 g/cm

^{3}) (1000 cm^{3}) = 1140 g1140 g / 2.04 g/mol = 558.824 mol

You could use any volume you want. I used 1.00 L to make the PV = nRT calculation just a tiny bit simpler.

2) Use the Ideal Gas Law:

PV = nRT(2.01 x 10

^{9}atm) (1.00 L) = (558.824 mol) (0.08206 L atm / mol K) (T)T = 4.38 x 10

^{7}K (to three sig figs)

**Bonus Example #2:** (a) A gas has a temperature of 300. K and a pressure of 104 kPa. Find the volume occupied by 1.05 mol of this gas, assuming it is ideal. (b) Assuming the gas molecules can be approximated as small spheres of diameter 3.0 x 10¯^{10} m , determine the fraction of the volume found in part (a) that is occupied by the molecules.

**Solution:**

1) PV = nRT:

(104 kPa / 101.325 kPa/atm) (V) = (1.05 mol) (0.08206 L atm / mol K) (300. K)V = 25.184 L

To three sig figs, the answer is 25.2 L

2) PV = nRT with slightly different numbers:

PV = nRT whereP = 104000 Pa, n = 1.05 mol, R = 8.314 Pa m

^{3}/mol K, T = 300. K.V = nRT/p = (315) (8.31447 m

^{3}) / 104000 = 0.025183 m^{3}<--- note the 315, which comes from 300. x 1.05Note how only the final unit that survives in the answer is shown.

V = 0.0252 m

^{3}

3) The number of molecules in 1.05 moles:

(1.05 mol) (6.022 x 10^{23}molecules mol¯^{1}) = 6.3231 x 10^{23}molecules

4) The volume of an individual molecule is:

(4/3) (3.14159) (1.5 x 10¯^{10}m)^{3}= 1.4137 x 10¯^{29}m^{3}

5) The total volume of the molecules in m^{3}:

(6.3231 x 10^{23}molecules) (1.4137 x 10¯^{29}m^{3}/ molecule) = 0.00000894 m^{3}

6) The fraction of the gas volume that is occupied by matter is:

0.00000894 m^{3}/ 0.025183 m^{3}= 0.000355 or about 1/2817 of the total volume