The heating curve for a sample of pure ethanol is provided above. [VISUAL] The temperature was recor — Thermodynamics Chemistry Question

STEPS:

1. Identify the phases represented by segments R and T on the heating curve:
* A heating curve plots temperature on the vertical y-axis against heat added on the horizontal x-axis.
* The diagonal segments correspond to the heating of a single, stable phase of matter, while the flat horizontal plateaus represent phase transitions occurring at a constant temperature.
* Since the sample starts as a solid at $100\text{ K}$ and is heated continuously:
* Segment P is heating the solid.
* Segment Q is the solid-to-liquid phase change (melting/fusion).
* Segment R is heating the liquid phase.
* Segment S is the liquid-to-gas phase change (boiling/vaporization).
* Segment T is heating the gas phase.
* Therefore, segment R represents liquid ethanol and segment T represents gaseous ethanol.
2. Recall the thermodynamic relationship for heating a single phase:
* The quantity of heat ($q$) absorbed by a substance to change its temperature ($\Delta T$) is given by the calorimetry formula:
$$q = mc\Delta T$$
where $m$ is the mass of the substance and $c$ is its specific heat capacity.
3. Analyze the mathematical definition of the slope on this heating curve:
* Rearranging the calorimetry equation to express the temperature change ($\Delta T$) as a function of heat added ($q$) yields:
$$\Delta T = \left(\frac{1}{mc}\right)q$$
* Since the y-axis is temperature ($T$) and the x-axis is heat added ($q$), the slope of a heating segment is defined as:
$$\text{Slope} = \frac{\Delta T}{q} = \frac{1}{mc}$$
4. Relate the slope of each segment to the specific heat capacity ($c$):
* The mass ($m$) of the ethanol sample is constant at $50.0\text{ g}$.
* Consequently, the slope of a diagonal segment is inversely proportional to the specific heat capacity ($c$) of the phase being heated ($\text{Slope} \propto \frac{1}{c}$).
* A steeper (greater) slope means that less heat is required to raise the temperature of that phase by a given amount, which corresponds to a smaller specific heat capacity.
5. Compare the slopes to find the correct relationship:
* The problem states that the slope of segment T is greater than the slope of segment R:
$$\text{Slope}_T > \text{Slope}_R \implies \frac{1}{m \cdot c_{\text{gas}}} > \frac{1}{m \cdot c_{\text{liquid}}} \implies c_{\text{gas}} < c_{\text{liquid}}$$
* This proves that the specific heat capacity of the gaseous ethanol is less than the specific heat capacity of liquid ethanol, confirming Option A as the correct answer.

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WHY_OTHERS_WRONG:

• Option B is incorrect: This option asserts that gaseous ethanol has a *greater* specific heat capacity than liquid ethanol. If that were true, the gas phase would require more heat input per degree of temperature change, which would make its heating line (segment T) less steep (smaller slope) than segment R. This directly contradicts both the visual curve and the question premise.
• Options C and D are incorrect: The heat of fusion ($\Delta H_{\text{fus}}$) and heat of vaporization ($\Delta H_{\text{vap}}$) represent the energy needed to complete phase changes. These constant-temperature processes are represented on the heating curve by the flat, horizontal plateaus (segments Q and S, respectively) where the slope is zero. Because segments R and T represent temperature changes within a single phase rather than phase transitions, their slopes are entirely independent of enthalpies of fusion or vaporization.