Question

Consider a reversible thermodynamic process involving one mole of ideal gas represented by a line on the $P-V$ diagram connecting two states X (4 bar, 1 L, $\text{T}_1$) and Y (1 bar, 2 L, $\text{T}_2$) as shown in the figure:

During the process the change in temperature (in K) as a function of volume (in L) is best represented as:

• A. A parabolic curve opening downwards with a maximum temperature.
• B. A hyperbolic curve decreasing monotonically.
• C. A straight line with a negative slope.
• D. A parabolic curve opening upwards with a minimum temperature.

Hint:

Any straight-line process on a $P-V$ diagram with a negative slope will always yield a quadratic relationship for temperature as a function of volume ($T \propto -V^2$), which manifests as a downward-opening parabola.

Answer 1

The Correct Option is A

Step 1: Set up the straight line in P and V.
The process goes in a straight line on the P-V graph from X (4 bar, 1 L) to Y (1 bar, 2 L). A straight line means $P = mV + c$. We find the slope and intercept from the two end points.

Step 2: Find the slope and intercept.
The slope is $m = \frac{1-4}{2-1} = -3$ bar per L. Using point X, $4 = (-3)(1) + c$, so $c = 7$. Thus $P = -3V + 7$.

Step 3: Bring in the gas law.
For one mole of ideal gas, $PV = RT$, so $T = \frac{PV}{R}$. We substitute the line equation for P to get T as a function of V alone.

Step 4: Substitute.
Plugging $P = -3V + 7$ gives \[ T = \frac{(-3V + 7)V}{R} = \frac{-3V^2 + 7V}{R}. \] This is a quadratic in V with a negative $V^2$ term.

Step 5: Read the shape.
A quadratic with a negative leading coefficient is a parabola that opens downward. A downward parabola has a single highest point, which is a maximum temperature somewhere along the path.

Step 6: Confirm the maximum lies on the path.
The peak of $T$ is at $V = \frac{7}{6} \approx 1.17$ L, which sits between 1 L and 2 L, so the maximum really occurs during the process. So T versus V is a downward parabola with a maximum temperature.

\[ \boxed{T(V) = \dfrac{-3V^2 + 7V}{R}:\ \text{a downward parabola with a maximum}} \]