Understanding Gas Laws
Gas laws describe the behavior of gases under varying conditions. To understand the Ideal Gas Equation ($PV=nRT$), we must first define its foundations.
Boyle's Law
Boyle's Law states that for a fixed amount of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional: $P \propto \frac{1}{V} \implies PV = k_1$
Charles's Law
Charles's Law states that the volume of a given mass of an ideal gas is directly proportional to its absolute temperature, provided the pressure remains constant: $V \propto T \implies \frac{V}{T} = k_2$
Deriving $PV = nRT$
By combining Boyle's Law, Charles's Law, and Avogadro's Law ($V \propto n$), we obtain: $V \propto \frac{nT}{P} \implies PV = nRT$ Where $R$ is the Universal Gas Constant.
Problem Solution: Ratio of Molecular Masses
Given:
- Gas A: $m_A = 2\text{g}$, $P_A = 1\text{ atm}$, $T = 25^{\circ}\text{C}$
- Mixture (A+B): $P_{total} = 1.5\text{ atm}$, $m_B = 3\text{g}$
- Since the flask and temperature are constant, $V$ and $RT$ are constant.
Step 1: Analyze Gas A $P_A V = n_A RT \implies 1 \cdot V = \left(\frac{2}{M_A}\right) RT$ --- (Eq. 1)
Step 2: Analyze Gas B Since total pressure $P_{total} = P_A + P_B$, then $P_B = P_{total} - P_A = 1.5 - 1 = 0.5\text{ atm}$. $P_B V = n_B RT \implies 0.5 \cdot V = \left(\frac{3}{M_B}\right) RT$ --- (Eq. 2)
Step 3: Calculate Ratio Divide Eq 1 by Eq 2: $\frac{1}{0.5} = \frac{2/M_A}{3/M_B} = \frac{2}{M_A} \cdot \frac{M_B}{3}$ $2 = \frac{2 M_B}{3 M_A} \implies \frac{M_A}{M_B} = \frac{2}{3 \cdot 2} = \frac{1}{3}$
The ratio of molecular masses $\frac{M_A}{M_B}$ is 1:3.