Question

Fundamental equation of state for ideal gas

• A. \(PV = nRT\)
• B. \(PV^2 = nRT\)
• C. \(\frac{P}{V} = nRT\)
• D. \(P + V = nRT\)

Hint:

Ensure your units are consistent when using the ideal gas law: \[ P \text{ (Pa)}, V \text{ (m}^3\text{)}, n \text{ (mol)}, T \text{ (Kelvin, K)}, R = 8.314\text{ J/(mol}\cdot\text{K)} \] Alternatively, you can use: \[ P \text{ (atm)}, V \text{ (L)}, n \text{ (mol)}, T \text{ (K)}, R = 0.0821\text{ L}\cdot\text{atm/(mol}\cdot\text{K)} \] Temperature must always be expressed on an absolute scale (Kelvin).

Answer 1

The Correct Option is A

Concept: An equation of state is a constitutive mathematical relationship that links state variables—specifically pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and particle or molar quantity (\(n\))—for a given substance under specified physical conditions. For an ideal gas, this relationship is derived by combining several empirical gas laws: Boyle's Law, Charles's Law, and Avogadro's Law. Detailed Derivation Steps:
• Boyle's Law: States that at a constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure: \[ V \propto \frac{1}{P} \quad (\text{at constant } T, n) \]
• Charles's Law: States that at a constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature: \[ V \propto T \quad (\text{at constant } P, n) \]
• Avogadro's Law: States that at a constant temperature and pressure, the volume of a gas is directly proportional to the number of moles present: \[ V \propto n \quad (\text{at constant } P, T) \] Combining these three proportional relations yields a single expression: \[ \text{Volume } (V) \propto \frac{n \cdot T}{P} \] To convert this proportionality into an equality, we introduce the universal gas constant, denoted as \(R\): \[ V = \frac{nRT}{P} \] Multiplying both sides by pressure (\(P\)) gives the classic form of the Ideal Gas Equation: \[ PV = nRT \] Options (2), (3), and (4) propose incorrect algebraic relationships that violate these fundamental physical laws. Thus, Option (1) is the correct choice.