Physics and Mathematics
Universal Gas Constant
1. Statement of the Concept
The Universal Gas Constant (R) is a fundamental constant that appears in the Ideal Gas Equation:
[pV = nRT]
It relates the pressure, volume, temperature, and amount of gas in moles, and remains the same for all ideal gases, hence called universal.
2. Clear Explanation
From the Ideal Gas Law,
[pV = nRT]
where
- [p] = pressure of the gas,
- [V] = volume of the gas,
- [n] = number of moles,
- [T] = absolute temperature, and
- [R] = universal gas constant.
For 1 mole of an ideal gas,
[pV = RT]
Thus,
[R = \dfrac{pV}{T}]
The constant [R] provides the proportionality between energy (from pressure and volume) and temperature for a given amount of gas.
It signifies the energy per mole per degree Kelvin.
3. Dimensions and Units
Dimensional formula of R:
[R = \dfrac{pV}{T}] [\Rightarrow R] = [M^1L^2T^{-2}\theta^{-1}]
Common units of R:
| System | Value of R | Unit |
|---|---|---|
| SI | [8.314] | [J mol^{-1} K^{-1}] |
| CGS | [8.314\times10^{7}] | [erg mol^{-1} K^{-1}] |
| Liter–atm | [0.0821] | [L atm mol^{-1} K^{-1}] |
| Calorie | [1.987] | [cal mol^{-1} K^{-1}] |
4. Key Features
- ( R ) is universal, i.e., same for all gases.
- It connects microscopic (molecular) and macroscopic (thermodynamic) behavior of gases.
- ( R ) relates to Boltzmann’s constant (k) by:
[R = N_A k]
where [N_A] = Avogadro’s number = [6.022\times10^{23},mol^{-1}]. - It provides the link between energy per molecule and energy per mole.
- It appears in many thermodynamic equations, including those for specific heats, entropy, and internal energy.
5. Important Formulas to Remember
| Formula | Description |
|---|---|
| [pV = nRT] | Ideal gas equation |
| [R = \dfrac{pV}{T}] | Definition of R |
| [R = N_A k] | Relation between universal and Boltzmann constant |
| [R = \dfrac{C_p – C_v}{1}] | Relation with specific heats (per mole basis) |
6. Conceptual Questions (with Solutions)
1. Why is R called “universal”?
Because its value is the same for all ideal gases, independent of their nature or molecular composition.
2. If 1 mole of an ideal gas occupies 22.4 L at 1 atm and 273 K, find R.
[R = \dfrac{pV}{T}] [= \dfrac{1\times22.4}{273}] [= 0.0821\,L\,atm\,mol^{-1}\,K^{-1}]
3. How is R connected to the energy of a gas?
[R] converts temperature (in K) to energy units through [pV = nRT], showing how thermal energy depends on temperature.
4. What is the SI dimensional formula of R?
[M^1L^2T^{-2}\theta^{-1}]
5. How does R relate to k (Boltzmann constant)?
[R = N_A k], linking molecular and molar levels of energy.
6. If pressure and temperature both double, what happens to R?
R remains constant because it is a universal constant, not dependent on conditions.
7. What is the energy equivalent of R in joules per mole per kelvin?
[8.314\,J\,mol^{-1}\,K^{-1}]
8. How can R be expressed in terms of specific heats?
[R = C_p – C_v]
9. Can R vary for real gases?
Theoretical R is constant; however, real gases may deviate due to intermolecular forces and finite molecular volume.
10. What is the relationship between R and the slope of an isothermal curve?
Using [pV = nRT], [p = \dfrac{nRT}{V}], the slope [\dfrac{dp}{dV}] shows inverse proportionality, but R itself remains fixed.
7. FAQ / Common Misconceptions
1. Is R different for different gases?
No. R is universal and has the same value for all ideal gases.
2. Does R depend on pressure or temperature?
No, it is a constant independent of external conditions.
3. Is R the same as Boltzmann’s constant?
No, [k] is per molecule, while [R = N_A k] is per mole.
4. Can R change for real gases?
For practical calculations under non-ideal conditions, deviations occur, but the *true universal value* remains constant.
5. Why are there different numerical values of R?
Because of different unit systems (SI, CGS, liter–atm, etc.).
6. Is R related to Avogadro’s law?
Yes, since the ideal gas law incorporates Avogadro’s principle that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.
7. Is R used in thermodynamics only?
No, R also appears in kinetic theory, statistical mechanics, and heat transfer.
8. Why is R used instead of k in macroscopic equations?
Because k applies to a single molecule, whereas R applies to one mole, making it suitable for bulk (macroscopic) systems.
9. What is meant by R connecting macroscopic and microscopic properties?
R links molecular energy (via k) to measurable properties like pressure, volume, and temperature.
10. Does R have the same dimensions as energy per temperature per mole?
Yes — it represents [Energy/(mol·K)].
8. Practice Questions (with Step-by-Step Solutions)
Q1. Find the value of R if 1 mole of gas occupies 22.4 L at 1 atm and 273 K.
Solution:
[R = \dfrac{pV}{T}] [= \dfrac{1\times22.4}{273}] [= 0.0821L atm mol^{-1},K^{-1}]
Q2. Convert this value of R to SI units.
Solution:
1 atm = [1.013\times10^5 Pa], 1 L = [10^{-3}m^3]
[R] [= 0.0821\times1.013\times10^5\times10^{-3}] [= 8.314 J mol^{-1} K^{-1}]
Q3. If [R = N_A k], find [k].
Solution:
[k] [= \dfrac{R}{N_A}] [= \dfrac{8.314}{6.022\times10^{23}}] [= 1.38\times10^{-23} J K^{-1}]
Q4. If 2 moles of gas occupy 0.05 m³ at 300 K, find its pressure.
Solution:
[p] [= \dfrac{nRT}{V}] [= \dfrac{2\times8.314\times300}{0.05}] [= 9.98\times10^4 Pa]
Q5. Show that [R = C_p – C_v] for one mole of gas.
Solution:
From first law and definitions of specific heats:
[C_p – C_v] [= (\dfrac{dQ}{dT})_p – (\dfrac{dQ}{dT})_v] [= R]
Hence proved.
Sign in with Google