Which of the following plot(s) is/are correct representation(s) of Boyle's Law?
Show Hint
Always formulate the explicit mathematical equation for each axis variable before verifying the graph shape:
- $y \propto x \rightarrow$ straight line through origin.
- $y \propto \frac{1}{x} \rightarrow$ rectangular hyperbola.
- $\log y = - \log x + c \rightarrow$ straight line with slope $-1$.
Step 1: Understanding the Concept:
Boyle's Law states that for a fixed amount of gas at constant temperature, the pressure ($P$) is inversely proportional to its volume ($V$). Mathematically:
\[ P \propto \frac{1}{V} \rightarrow PV = k \rightarrow P = \frac{k}{V} \] Step 2: Detailed Explanation:
- Statement (A): Correct. If we plot $P$ against $x$, where $x = 1/V$, the equation $P = kx$ represents a straight line passing through the origin with slope $k$.
- Statement (B): Correct. Taking the log of both sides of $PV = k$:
\[ \log P + \log V = \log k \]
\[ \log P = -1 \log V + \log k \]
This is a linear equation ($y = mx + c$) where the slope $m = -1$.
- Statement (C): Correct. The relationship $P = k/V$ is the mathematical definition of a rectangular hyperbola. As $V$ increases, $P$ decreases non-linearly.
- Statement (D): Incorrect. A straight line with a negative slope would imply a linear relationship $P = -mV + c$, which contradicts the inverse proportionality $PV = k$. Step 3: Final Answer:
Plots (A), (B), and (C) are correct representations.
