Which graph correctly represents the relationship between volume and temperature for an ideal gas at constant pressure and fixed mass?

Angular frequency ($\omega$) is the rate of change of phase. For a complete oscillation, the phase changes by $2\pi$ radians over a time period $T$.

The formula for angular frequency is given by:

Given that the change in phase is $2\pi$ and the time taken is $T$:

The calculated angular frequency matches option D.

The final answer is $\boxed{\text{D) } \omega = \frac{2\pi}{T}}$.

The question asks for the correct graph representing the relationship between volume ($V$) and temperature for an ideal gas at constant pressure and fixed mass. This describes Charles's Law.

Charles's Law states that for a fixed mass of gas at constant pressure, the volume ($V$) is directly proportional to its absolute temperature ($T_K$).

This can be written as $V = k T_K$, where $k$ is a constant.

Absolute temperature ($T_K$) is related to Celsius temperature ($t_C$) by the formula:

Substituting this into Charles's Law:

This equation shows a linear relationship between $V$ and $t_C$.

The graph of $V$ versus $t_C$ should be a straight line.
When $t_C = -273.15^\circ C$ (absolute zero), the volume $V$ becomes zero:

So, the straight line graph should extrapolate to $V=0$ at $t_C = -273.15^\circ C$.

• Option A shows a straight line graph of $V$ versus $t_C$ that extrapolates to $V=0$ at $-273^\circ C$. This is consistent with Charles's Law.
• Option B shows a curved graph, which is incorrect for the relationship between $V$ and $t_C$.

The final answer is $\boxed{\text{A}}$.