# Gas constant: what it is, calculation and examples

The **gas** constant is a physical constant that appears in various equations, the best known being that which links the four variables that characterize an ideal gas: pressure, volume , temperature, and the amount of matter .

The ideal gas is a hypothetical model of gases, in which the particles that compose it interact very little and are much smaller than the total volume occupied. In this case, the four variables mentioned follow the following simple equation, which results from combining the laws of Boyle, Charles and Avogadro:

P ∙ V = n ∙ R ∙ T

Where P is the pressure, V is the volume, T the temperature, n the number of moles present in a portion of ideal gas and R is precisely the gas constant. Its value, determined experimentally, is 0.0821 L ∙ atm / K ∙ mol.

The name of R for the constant is believed to be in honor of the French chemist Henri Victor Regnault (1810-1878), who worked extensively measuring the properties of gases.

The constant R can be expressed in different systems of units, and then its numerical value changes. For this reason, it is convenient to pay close attention to the system of units used when working and thus use the appropriate value of the constant.

**How to determine the gas constant**

Despite the simplicity of the ideal gas model, many gases behave this way when the temperature is 0º C (273.15 K) and the pressure is equivalent to 1 atmosphere , abbreviated as *1 atm* .

In that case, 1 mole of any gas occupies a volume of 22,414 L, just slightly more than that of a basketball. These pressure and temperature *conditions* are known as *standard conditions* .

If their values are substituted in the ideal gas equation of state P ∙ V = n ∙ R ∙ T and R is cleared, the following result is obtained:

It is common to check the value of the gas constant through simple experiments: for example, obtaining a portion of gas through a chemical reaction and measuring its pressure, volume and temperature.

**Units of the gas constant**

The quantities involved in the ideal gas model are usually measured in different units. The value given above is frequently used in calculations, but it is not the one corresponding to the SI International System of Units, which is the standard in science.

In this system of units, the *kelvin* is the unit of temperature, the pressure is measured in *pascals* (Pa) and the volume in *cubic meters* (m ^{3} ).

To write the gas constant in this system of units, the following conversion factors must be used, which relate atmospheres to pascals, and liters to cubic meters:

1L = 1 x 10 ^{-3} m ^{3}

1 atm = 101325 Pa

Note that 1 pascal = 1 newton / m ^{2} , so 1 Pa.m ^{3} = 1 newton ∙ m = 1 joule = 1 J. The joule is the unit for energy, and the gas constant relates energy to temperature and quantity of matter.

The calorie is a unit that is still frequently used to measure energy. The equivalence with the joule is:

1 calorie = 4.18 J

If you prefer to use the calorie instead of the joule, the gas constant is valid in this case:

R = 1.9872 cal / K ∙ mol

It is possible to combine various units of energy, temperature, and quantity of matter to express R

**Relationship with Boltzmann’s constant and Avogadro’s number**

In thermodynamics there are three important constants that are related: the gas constant R, the Boltzmann constant k _{B} and the Avograd number N _{A} :

R = N _{A} ∙ k _{B}

**Application exercises**

**Exercise 1**

It is desired to determine in the laboratory the value of the gas constant, for which a quantity of ammonium nitrate NH _{4} NO _{3} is thermally decomposed and nitrous oxide N _{2} O is obtained , a gas known for its anesthetic effect, in addition to Water.

From this experiment 0.340 L of nitrous oxide were obtained, equivalent to 0.580 g of gas, at a pressure of 718 mmHg and a temperature of 24ºC. Determine what R is worth in this case, assuming that nitrous oxide behaves like an ideal gas.

**Solution**

Millimeters of mercury are also units for measuring pressure. In this case, the gas constant is expressed in terms of another set of units. Regarding the mass in grams, it can be converted to moles through the nitrous oxide formula, consulting the atomic mass of nitrogen and oxygen in tables:

-Nitrogen: 14.0067 g / mol

-Oxygen: 15.9994 g / mol

Therefore 1 mole of nitrous oxide has:

(2 x 14.0067 g / mol) + 15.9994 g / mol = 44.0128 g / mol

Now convert the number of grams of nitrous oxide to moles:

0.580 g = 0.580 gx 1mol /44.0128 g = 0.013178 mol

On the other hand, 24 ºC is equivalent to 297.17 K, in this way:

In this set of units, the value of the gas constant under standard conditions, according to the tables, is R = 62.36365 mmHg ∙ L / K ∙ mol. Can the reader make a guess about the reason for this small difference?

**Exercise 2**

Where P and Po represent, respectively, the pressure at altitude h and sea level, g is the familiar value of the acceleration of gravity, M is the average molar mass of air, R is the gas constant, and T the temperature. .

It is asked to find the atmospheric pressure at a height h = 5 km, assuming that the temperature remains at 5ºC.

**Data:**

g = 9.8 m / s ^{2}

M = 29.0 g / mol = 29.0 x 10 ^{-3} kg / mol

R = 8.314 J / K ∙ mol

P _{o} = 1 atm

**Solution**

The values are substituted, taking care to maintain the homogeneity of the units in the exponential argument. Since the value of the acceleration due to gravity is known in SI units, the argument (which is dimensionless) works in these units:

h = 5 km = 5000m

T = 5 ºC = 278.15 K

-gMh / RT = (- 9.8 x 29.0 x 10 ^{-3} x 5000) / (8.314 J / K ∙ mol x 278.15 K) = -0.6144761

e ^{-0.6144761} = 0.541

Therefore:

P = 0.541 x 1 atm = 0.541 atm

Conclusion: the atmospheric pressure is reduced almost to half its value at sea level when the height is 5 km (Everest has a height of 8,848 km).