Question

which formula represents gay-lussac’s law?
\\(\frac{v_1}{t_1} = \frac{v_2}{t_2}\\)
\\(\frac{p_1}{t_1} = \frac{p_2}{t_2}\\)
\\(p_1v_1 = p_2v_2\\)
\\(p_1t_1 = p_2t_2\\)

Explanation:

Step1: Recall Gay-Lussac's Law

Gay - Lussac's Law states that for a given mass of a gas at constant volume, the pressure of the gas is directly proportional to its absolute temperature. Mathematically, this can be written as $\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}$ (when volume is constant, $V_1 = V_2$ and from the ideal gas law $PV = nRT$, if $V$ and $n$ are constant, $P\propto T$).

Step2: Analyze each option

• Option 1: $\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}$ is Charles's Law (volume - temperature relationship at constant pressure).
• Option 2: $\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}$ matches the relationship from Gay - Lussac's Law (pressure - temperature relationship at constant volume).
• Option 3: $P_{1}V_{1}=P_{2}V_{2}$ is Boyle's Law (pressure - volume relationship at constant temperature).
• Option 4: $P_{1}T_{1}=P_{2}T_{2}$ does not represent any of the gas laws (the correct relationship for Gay - Lussac's is a direct proportion between $P$ and $T$ at constant $V$, not a product relationship).

Answer:

$\boldsymbol{\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}}$ (the second option)