How to Derive the Gas Constant Dimensions—Step-by-Step Guide
The universal gas constant is an important physical constant that appears in the ideal gas law and other thermodynamic equations. Its dimensional formula provides insight into the fundamental relationships among physical quantities such as mass, length, time, temperature, and amount of substance.
Definition and Role of Gas Constant
The universal gas constant, denoted as $R$, relates pressure, volume, temperature, and the number of moles of an ideal gas in the equation $PV = nRT$. This relationship forms the basis of the ideal gas law in thermodynamics.
Expression for Dimensions of Gas Constant
To derive the dimensions of the universal gas constant, start from the equation $R = \dfrac{PV}{nT}$, where $P$ is pressure, $V$ is volume, $n$ is the number of moles, and $T$ is absolute temperature.
Pressure has the dimensional formula $[ML^{-1}T^{-2}]$. Volume is expressed as $[L^{3}]$. Temperature is a fundamental quantity, denoted as $[\Theta]$ or $[K]$, and the number of moles is a countable quantity with no dimensions. Thus, dimensions for $n$ are $[mol]$.
Stepwise Dimensional Analysis
Substituting all quantities into the formula for $R$, the dimensional expression becomes:
$[R] = \dfrac{[ML^{-1}T^{-2}][L^3]}{[mol][\Theta]}$
Multiplying and simplifying the exponents yields:
$[R] = \dfrac{[ML^{2}T^{-2}]}{[mol][\Theta]} = [ML^{2}T^{-2}mol^{-1}\Theta^{-1}]$
Dimensional Formula and SI Unit of Gas Constant
The universal gas constant thus has the dimensional formula $[M^{1}L^{2}T^{-2}mol^{-1}\Theta^{-1}]$. The SI unit of $R$ is joule per mole per kelvin ($\mathrm{J\,mol^{-1}\,K^{-1}}$).
| Quantity | Dimensional Formula |
|---|---|
| Universal Gas Constant ($R$) | $[ML^2T^{-2}mol^{-1}\Theta^{-1}]$ |
The dimensions indicate that the gas constant represents the amount of energy per mole per kelvin of temperature change. For practice problems involving dimensions, refer to the Units and Measurements Mock Test 3 page.
Comparison with Related Physical Quantities
Other dimensionally similar quantities include the Boltzmann constant, which differs by replacing the mole with the particle. Universal gas constant and Boltzmann constant both express energy per degree of temperature, but their context varies.
The units of the gas constant are derived from the units of energy, amount of substance, and temperature, emphasizing its thermodynamic significance. For more dimensional analysis, see the Dimensions of Electric Flux resource.
Summary of Key Points
- Universal gas constant relates energy, mole, and temperature
- Dimensional formula: $[ML^2T^{-2}mol^{-1}\Theta^{-1}]$
- SI unit: $\mathrm{J\,mol^{-1}\,K^{-1}}$
- Derived from $PV = nRT$ ideal gas equation
A strong understanding of dimensional analysis for constants like $R$ supports error-checking in formulae and helps in solving advanced physics problems. Additional dimensional formulae can be studied on the Dimensions of Magnetic Flux and Dimensions of Volume pages.
FAQs on What Are the Dimensions of the Gas Constant (R)?
1. What are the dimensions of the gas constant?
The dimensions of the gas constant (R) in the SI unit system are kg m2 s-2 K-1 mol-1. In dimensional terms, this is represented as [M1 L2 T-2 K-1 N-1], where M = mass, L = length, T = time, K = temperature, and N = amount of substance (mole).
- Mass (M): 1
- Length (L): 2
- Time (T): -2
- Temperature (K): -1
- Mole (N or n): -1
2. What is the SI unit of the universal gas constant?
The SI unit of the universal gas constant (R) is Joule per mole per Kelvin (J mol-1 K-1).
- 1 Joule (J) = 1 kg m2 s-2
- Per mole (mol-1)
- Per Kelvin (K-1)
3. How do you derive the dimensional formula of the gas constant?
The dimensional formula of the gas constant (R) is derived from the ideal gas equation PV = nRT.
- P = pressure (ML-1T-2)
- V = volume (L3)
- n = amount (mol)
- T = temperature (K)
Dimensions of R: [M1L2T-2K-1N-1].
4. What is the value of the gas constant R in SI units?
The value of the gas constant (R) in SI units is 8.314 J mol-1 K-1.
- R = 8.314 Joule per mole per Kelvin
5. What are the applications of the gas constant in chemistry?
The gas constant (R) is widely used in chemistry for calculations and equations involving gases.
- In the ideal gas law (PV = nRT)
- Calculating molar volume of gases
- Relating energy, work, and temperature in thermodynamics
- In Arrhenius equation for reaction rates
6. Is the gas constant R the same in all unit systems?
The numerical value of R changes depending on the unit system, but its physical significance remains the same.
- In SI: 8.314 J mol-1 K-1
- In cgs: 8.314 × 107 erg mol-1 K-1
- In L·atm: 0.0821 L atm mol-1 K-1
7. Why is the gas constant important in the ideal gas law?
The gas constant (R) is a proportionality factor in the ideal gas law (PV = nRT), ensuring a relationship between physical quantities.
- It connects pressure, volume, temperature, and amount (n) of gas.
- Ensures accurate predictions of gas behavior under ideal conditions.
8. What is the dimensional formula of the universal gas constant?
The dimensional formula of the universal gas constant (R) is [M1L2T-2K-1N-1].
This is derived from its units (J mol-1 K-1) and is key in dimensional analysis of thermodynamic equations.
9. What is the relationship between Boltzmann constant and gas constant?
The gas constant (R) is related to the Boltzmann constant (k) by the equation R = NA × k.
- R = gas constant
- NA = Avogadro constant (number of molecules in a mole)
- k = Boltzmann constant
10. Can you give examples of numerical problems using the gas constant?
Numerical problems involving the gas constant (R) typically use the ideal gas law:
- Example: Calculate the volume occupied by 2 moles of a gas at 273 K and 1 atm.
Solution: Use PV = nRT, where R = 0.0821 L atm mol-1 K-1 - Other examples include finding pressure, volume, temperature or moles using the values of R in different units.
