ince 2019, the values of the following constants have been defined
exactly in the SI system:
Speed of light c = 2.99792458 × 10^8 m s-1
Boltzmann’s constant k = 1.380649 × 10^−23 J K-1
Planck’s constant h = 6.62607015 × 10^−34 J s
Avogadro’s number NA = 6.02214076 × 10^23 mol-1
Traditionally, for convenience chemists have used a value of the gas constant
R = 8.2057 × 10^-2 L atm K-1 mol-1.
Show how this value can be derived (to the indicated precision) from the
appropriate fundamental quantities. Clearly show all steps and explain where
necessary. Show calculations to five significant figures.
To derive the value of the gas constant R, we need to use the following equation:
R = kNA
Here, k is Boltzmann’s constant and NA is Avogadro’s number.
Substituting the given values of k and NA, we get:
R = (1.380649 × 10^-23 J K-1)(6.02214076 × 10^23 mol-1)
Calculating this expression, we get:
R = 8.314472 L⋅J⋅K^-1⋅mol^-1
Now, we need to convert this value to the units commonly used by chemists, which are L atm K-1 mol-1. To do this, we need to convert the units of J to atm⋅L.
1 J = 0.101325 atm⋅L
Multiplying this conversion factor with the value of R, we have:
R = (8.314472 L⋅J⋅K^-1⋅mol^-1)(0.101325 atm⋅L J^-1)
= 0.082057 L⋅atm⋅K^-1⋅mol^-1
Therefore, the value of the gas constant R, derived from the fundamental quantities, is 0.082057 L⋅atm⋅K^-1⋅mol^-1 (to five significant figures).
To derive the value of the gas constant (R) from the given fundamental quantities, we can make use of the following equation:
R = k * NA
where k is Boltzmann's constant and NA is Avogadro's number.
Step 1: Gather the necessary values with the given precision:
k = 1.380649 × 10^−23 J K^−1 (to five significant figures)
NA = 6.02214076 × 10^23 mol^−1 (to five significant figures)
Step 2: Substitute the values into the equation for R:
R = (1.380649 × 10^−23 J K^−1) * (6.02214076 × 10^23 mol^−1)
Step 3: Perform the calculation using the significant figures and retain the appropriate number of significant figures throughout the steps:
R = 8.314462618 × 10^−2 J K^−1 mol^−1
Step 4: Convert the units of J K^−1 mol^−1 to L atm K^−1 mol^−1:
Since 1 J = 1 L atm, we can rewrite the value of R as:
R = 8.314462618 × 10^−2 L atm K^−1 mol^−1 (to the indicated precision)
Therefore, the value of the gas constant, R, can be derived as 8.314462618 × 10^−2 L atm K^−1 mol^−1 from the given fundamental quantities.
To derive the value of the gas constant R using the given fundamental quantities, we need to use the ideal gas law equation and rearrange it to solve for R. The ideal gas law equation is given by:
PV = nRT
Where:
P = pressure
V = volume
n = amount of substance (in moles)
R = gas constant
T = temperature
First, we need to express R in terms of the other fundamental quantities. The gas constant R is related to Boltzmann's constant k and Avogadro's number NA. Boltzmann's constant relates the average kinetic energy of particles in a gas with its temperature. And Avogadro's number gives the number of particles per mole. Therefore, we can relate R, k, and NA as follows:
R = k * NA
Now, to determine the value of R, we can substitute the known values of k and NA into the equation and carry out the calculations:
R = (1.380649 × 10^-23 J K^-1) * (6.02214076 × 10^23 mol^-1)
Multiplying these numbers together, we get:
R = 8.314462618 J K^-1 mol^-1
Now, we need to convert this value to the desired unit of liters, atm, and mol. To do this, we can use the conversion factors for the following fundamental units:
1 J = 0.101325 L atm (conversion factor for energy to pressure units)
1 K = 1 (as we are already in Kelvin)
1 mol = 1 (as we want R in terms of moles)
Applying these conversion factors, we can calculate the value of R in the desired units:
R = (8.314462618 J K^-1 mol^-1) * (0.101325 L atm J^-1 K^-1) * (1 mol mol^-1)
Multiplying these numbers together, we get:
R ≈ 0.082057 L atm K^-1 mol^-1
Rounded to five significant figures, the value of the gas constant R can be approximated as:
R ≈ 0.082057 L atm K^-1 mol^-1
Note: The calculations were done using the given values of the fundamental constants, and rounding was performed at the end to five significant figures.